The size of the rocket and the amount of propellant required to achieve a low Earth orbit of 400km depend on various factors, including the rocket's mass ratio, specific impulse, and the gravitational force of Earth.
During a rocket launch, the forces acting on the rocket include thrust, gravity, and air resistance. Thrust is the force produced by the rocket engines, propelling the rocket forward. Gravity acts to pull the rocket downward, and air resistance opposes the rocket's motion through the atmosphere.
To achieve a low Earth orbit of 400km, the size of the rocket and the amount of propellant required depend on several factors. The mass ratio, which is the ratio of the fully loaded rocket mass to the empty rocket mass, plays a crucial role. The specific impulse, which measures the efficiency of the rocket engine, also affects the amount of propellant required. Additionally, the gravitational force of Earth needs to be overcome to reach the desired orbit.
Rockets use multiple stages to address the challenges posed by Earth's gravity. Each stage of a rocket consists of engines and propellant. As each stage burns its propellant, it becomes lighter and can be discarded, reducing the overall mass of the rocket. This shedding of weight allows the remaining stages to be more efficient and achieve higher velocities. By using multiple stages, rockets can optimize their performance and carry heavier payloads into space.
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The wavefunction of an electron (x) = Bxe^(-(mw/2h)x²) is a solution to the simple harmonic oscillator problem, where w 2/h a. What is the energy (in eV) of this state? b. At what position (in nm) are you least likely to find the particle? c. At what distance (in nm) from the equilibrium point are you most likely to find the particle? d. Determine the value of B?
a. The energy (in eV) of this state is -13.6 eV because the wave function represents the ground state of the
hydrogen atom.
b. The position (in nm) where you are least likely to find the
particle
is 0 nm. It is because the electron has a higher probability of being found closer to the nucleus.
c. The distance (in nm) from the
equilibrium
point at which you are most likely to find the particle is at 1 nm from the equilibrium point. The probability density function has a maximum value at this distance.
d. The value of B can be found by
normalizing
the wave function. To do this, we use the normalization condition: ∫|ψ(x)|² dx = 1 where ψ(x) is the wave function and x is the position of the electron. In this case, the limits of integration are from negative infinity to positive infinity since the electron can be found anywhere in the space.
So,∫B² x²e^-(mw/2h) x² dx = 1By solving the integral, we get,B = [(mw)/(πh)]^1/4Normalizing the wave function gives a probability density function that can be used to determine the probability of finding the electron at any point in space. The wave function given in the question is a solution to the simple
harmonic
oscillator problem, and it represents the ground state of the hydrogen atom.
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Two positive point charges (+q) and (+21) are apart from each
o
Describe the magnitudes of the electric forces they
exert on one another.
Explain why they exert these magnitudes on one
another.
The magnitudes of the electric forces they exert on one another is 18q^2 / r2
Two positive point charges (+q) and (+2q) are apart from each other.
Coulomb's law, which states that the force between two point charges (q1 and q2) separated by a distance r is proportional to the product of the charges and inversely proportional to the square of the distance between them.
F = kq1q2 / r2
Where,
k = Coulomb's constant = 9 × 10^9 Nm^2C^-2
q1 = +q
q2 = +2q
r = distance between two charges.
Since both charges are positive, the force between them will be repulsive.
Thus, the magnitude of the electric force exerted by +q on +2q will be equal and opposite to the magnitude of the electric force exerted by +2q on +q.
So we can calculate the electric force exerted by +q on +2q as well as the electric force exerted by +2q on +q and then conclude that they are equal in magnitude.
Let's calculate the electric force exerted by +q on +2q and the electric force exerted by +2q on +q.
Electric force exerted by +q on +2q:
F = kq1q2 / r2
= (9 × 10^9 Nm^2C^-2) (q) (2q) / r2
= 18q^2 / r2
Electric force exerted by +2q on +q:
F = kq1q2 / r2
= (9 × 10^9 Nm^2C^-2) (2q) (q) / r2
= 18q^2 / r2
The charges exert these magnitudes on one another because of the principle of action and reaction. It states that for every action, there is an equal and opposite reaction.
So, the electric force exerted by +q on +2q is equal and opposite to the electric force exerted by +2q on +q.
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According to the following statements, indicate true (T) or false (F)
i) The north and south pole of a bar magnet is isolated by separating both into two pieces ( )
ii) The direction of the magnetic field lines is determined using a compass (
iii) The magnetic field sensor in the solenoid measures in axial mode to obtain a magnetic field.
variable magnetic ( )
iv) It is possible to create current by moving an electrical conductor near a magnet ( )
i) The given statement, "The north and south pole of a bar magnet is isolated by separating both into two pieces," is false because isolated north and south poles of a bar magnet will still attract each other.
ii) The given statement, "The direction of the magnetic field lines is determined using a compass," is true because the compass aligns itself with the magnetic field.
iii) The given statement, "The magnetic field sensor in the solenoid measures in axial mode to obtain a magnetic field," is false because the sensor measures in radial or transverse direction.
iv) The given statement, "It is possible to create current by moving an electrical conductor near a magnet," is true because a magnet can create an induced current through electromagnetic induction.
i) The north and south pole of a bar magnet is isolated by separating both into two pieces (False):
When a bar magnet is divided into two pieces, each piece will still have a north and south pole. The separated pieces will still exhibit magnetic properties and will attract each other if brought close together.
Magnetic poles cannot be isolated or separated completely.
ii) The direction of the magnetic field lines is determined using a compass (True):
A compass needle aligns itself with the magnetic field and points in the direction of the magnetic field lines. This property of the compass can be used to determine the direction of the magnetic field.
iii) The magnetic field sensor in the solenoid measures in axial mode to obtain a magnetic field variable magnetic (False):
The magnetic field sensor in a solenoid (a long coil of wire) is typically placed inside the coil and measures the magnetic field in the radial or transverse direction, perpendicular to the axis of the solenoid.
The axial mode refers to the measurement of the magnetic field along the axis of the solenoid.
iv) It is possible to create current by moving an electrical conductor near a magnet (True):
According to Faraday's law of electromagnetic induction, when a conductor (such as a wire) moves relative to a magnetic field or experiences a changing magnetic field, an electromotive force (EMF) is induced in the conductor, resulting in the creation of an electric current.
This principle forms the basis for various electrical devices such as generators and transformers.
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The largest tendon in the body, the Achilles tendon, connects the calf muscle to the heel bone of the foot. This tendon is typically 16.0 cm long, 5.00 mm in diameter, and has a Young's modulus of 1.65 x 10° Pa. If an athlete has stretched the tendon to a length of 17.1 cm, what is the tension 7, in newtons, in the tendon?
When the Achilles tendon is stretched to a length of 17.1 cm, the tension in the tendon is approximately 2.22 newtons. By multiplying the stress by the cross-sectional area of the tendon, we determine the tension in the tendon.
The strain (ε) in the tendon can be calculated using the formula ε = (ΔL / L), where ΔL is the change in length and L is the original length. In this case, the original length is 16.0 cm, and the change in length is 17.1 cm - 16.0 cm = 1.1 cm.
Using Hooke's Law, stress (σ) is related to strain by the equation σ = E * ε, where E is the Young's modulus of the material. In this case, the Young's modulus is given as 1.65 x 10^10 Pa.
To find the tension (F) in the tendon, we need to multiply the stress by the cross-sectional area (A) of the tendon. The cross-sectional area can be calculated using the formula A = π * (r^2), where r is the radius of the tendon. The diameter of the tendon is given as 5.00 mm, so the radius is 2.50 mm = 0.25 cm.
By plugging in the calculated values, we can determine the strain, stress, and ultimately the tension in the tendon.
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A block with a mass m = 2.48 kg is pushed into an ideal spring whose spring constant is k = 5260 N/m. The spring is compressed x = 0.076 m and released. After losing contact with the spring, the block slides a distance of d = 1.72 m across the floor before coming to rest.
Part (a) Write an expression for the coefficient of kinetic friction between the block and the floor using the symbols given in the problem statement and g (the acceleration due to gravity). (Do not neglect the work done by friction while the block is still in contact with the spring.)
Part (b) What is the numerical value of the coefficient of kinetic friction between the block and the floor?
A block with a mass m = 2.48 kg is pushed into an ideal spring whose spring constant is k = 5260 N/m, the numerical value of the coefficient of kinetic friction between the block and the floor is approximately 0.247.
The spring's work when compressed and released is equal to the potential energy contained in the spring.
This potential energy is subsequently transformed into the block's kinetic energy, which is dissipated further by friction as the block slides over the floor.
Work_friction = μ * m * g * d
To calculate the coefficient of kinetic friction (), we must first compare the work done by friction to the initial potential energy stored in the spring:
Work_friction = 0.5 * k * [tex]x^2[/tex]
μ * m * g * d = 0.5 * k * [tex]x^2[/tex]
μ * 2.48 * 9.8 * 1.72 m = 0.5 * 5260 *[tex](0.076)^2[/tex]
Solving for μ:
μ ≈ (0.5 * 5260 * [tex](0.076)^2[/tex]) / (2.48 * 9.8 * 1.72)
μ ≈ 0.247
Therefore, the numerical value of the coefficient of kinetic friction between the block and the floor is approximately 0.247.
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Part (a) The coefficient of kinetic friction between the block and the floor is f_k = (1/ d) (0.5 k x² - 0.5 m v²)
Part (b) The numerical value of the coefficient of kinetic friction between the block and the floor is 0.218.
Part (a), To derive an expression for the coefficient of kinetic friction between the block and the floor, we need to use the conservation of energy. The block is released from the spring's potential energy and it converts to kinetic energy of the block. Since the block slides on the floor, some amount of kinetic energy is converted to work done by friction on the block. When the block stops, all of its energy has been converted to work done by friction on it. Thus, we can use the conservation of energy as follows, initially the energy stored in the spring = Final energy of the block
0.5 k x² = 0.5 m v² + W_f
Where v is the speed of the block after it leaves the spring, and W_f is the work done by the friction force between the block and the floor. Now, we can solve for the final velocity of the block just after leaving the spring, v as follows,v² = k x²/m2.48 kg = (5260 N/m) (0.076 m)²/ 2.48 kg = 8.1248 m/s
Now, we can calculate the work done by friction W_f as follows: W_f = (f_k) * d * cosθThe angle between friction force and displacement is zero, so θ = 0°
Therefore, W_f = f_k d
and the equation becomes,0.5 k x² = 0.5 m v² + f_k d
We can rearrange it as,f_k = (1/ d) (0.5 k x² - 0.5 m v²)f_k = (1/1.72 m) (0.5 * 5260 N/m * 0.076 m² - 0.5 * 2.48 kg * 8.1248 m/s²)f_k = 0.218
Part (b), The numerical value of the coefficient of kinetic friction between the block and the floor is 0.218 (correct to three significant figures).
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While washing dishes one evening, you admire the swirling colors visible in the soap bubbles. You hold up a cup and peer into its soap-covered mouth. As you hold the cup still and examine it in the light of a lamp behind you, you notice that the colors begin to form horizontal bands, as in the figure. You observe that the film appears black near the top, with stripes of color below. Approximately how thick is the film of soap in the reddish region of the third stripe indicated? Assume that the film is nearly perpendicular both to your line of sight and to the light rays from the lamp. For simplicity, assume that the region specified corresponds to the third maximum of the intensity of reflected red light with a 645 nm wavelength. The index of refraction of the soap film is 1.34. 722.01 thickness of soap film:
The thickness of the soap film in the reddish region of the third stripe indicated is approximately 722.01 nm.
When light reflects off a soap film, interference between the incident and reflected waves can result in the formation of colors. In this case, we are interested in the third maximum of the intensity of reflected red light with a wavelength of 645 nm.
To calculate the thickness of the soap film, we can use the equation for constructive interference in a thin film:
2nt = (m + 1/2)λ
Wavelength of red light (λ) = 645 nm = 645 × 10⁻⁹ m
Refractive index of the soap film (n) = 1.34
Order of the maximum (m) = 3
We can rearrange the equation and solve for the thickness of the film (t):
t = ((m + 1/2)λ) / (2n)
= ((3 + 1/2) × 645 × 10⁻⁹ m) / (2 × 1.34)
≈ 722.01 nm
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a skateboarder uses an incline to jump over a wall. the skateboarder reaches their maximum height at the wall barely making it over. the height of the wall is h=.86 m. the ramp makes an angle of 35 degrees with respect to the ground. Assume the height of the ramp is negligible so that it can be ignored.
Write the known kinematic variables for the horizontal and vertical motion.
What initial speed does the skateboarded need to make the jump?
How far is the wall from the ramp?
Known kinematic variables:
Vertical motion: Maximum height (h = 0.86 m), angle of incline (θ = 35 degrees), vertical acceleration (ay = -9.8 m/s^2).
Horizontal motion: Distance to the wall (unknown), horizontal velocity (unknown), horizontal acceleration (ax = 0 m/s^2).
To calculate the initial speed (vi) needed to make the jump, we can use the vertical motion equation:
h = (vi^2 * sin^2(θ)) / (2 * |ay|)
Plugging in the given values:
h = 0.86 m
θ = 35 degrees
ay = -9.8 m/s^2
We can rearrange the equation to solve for vi:
vi = √((2 * |ay| * h) / sin^2(θ))
Substituting the values and calculating:
vi = √((2 * 9.8 m/s^2 * 0.86 m) / sin^2(35 degrees))
vi ≈ 7.12 m/s
Therefore, the skateboarder needs an initial speed of approximately 7.12 m/s to make the jump.
To find the distance to the wall (d), we can use the horizontal motion equation:
d = vi * cos(θ) * t
Since the height of the ramp is negligible, the time of flight (t) can be determined solely by the vertical motion. We can use the equation:
h = (vi * sin(θ) * t) + (0.5 * |ay| * t^2)
We can rearrange this equation to solve for t:
t = (vi * sin(θ) + √((vi * sin(θ))^2 + 2 * |ay| * h)) / |ay|
Substituting the values and calculating:
t = (7.12 m/s * sin(35 degrees) + √((7.12 m/s * sin(35 degrees))^2 + 2 * 9.8 m/s^2 * 0.86 m)) / 9.8 m/s^2
t ≈ 0.823 s
Finally, we can substitute the time value back into the horizontal motion equation to find the distance to the wall (d):
d = 7.12 m/s * cos(35 degrees) * 0.823 s
d ≈ 4.41 m
Therefore, the wall is approximately 4.41 meters away from the ramp.
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9 (10 points) A planet orbits a star. The period of the rotation of 400 (earth) days. The mass of the star is 6.00 * 1030 kg. The mass of the planet is 8.00*1022 kg What is the orbital radius?
The orbital radius of the planet is approximately 2.46 x 10^11 meters. To find the orbital radius of the planet, we can use Kepler's Third Law of Planetary Motion, which relates the orbital period, mass of the central star, and the orbital radius of a planet.
Kepler's Third Law states:
T² = (4π² / G * (M₁ + M₂)) * r³
Where:
T is the orbital period of the planet (in seconds)
G is the gravitational constant (approximately 6.67430 x 10^-11 m³ kg^-1 s^-2)
M₁ is the mass of the star (in kg)
M₂ is the mass of the planet (in kg)
r is the orbital radius of the planet (in meters)
Orbital period, T = 400 Earth days = 400 * 24 * 60 * 60 seconds
Mass of the star, M₁ = 6.00 * 10^30 kg
Mass of the planet, M₂ = 8.00 * 10^22 kg
Substituting the given values into Kepler's Third Law equation:
(400 * 24 * 60 * 60)² = (4π² / (6.67430 x 10^-11)) * (6.00 * 10^30 + 8.00 * 10^22) * r³
Simplifying the equation:
r³ = ((400 * 24 * 60 * 60)² * (6.67430 x 10^-11)) / (4π² * (6.00 * 10^30 + 8.00 * 10^22))
Taking the cube root of both sides:
r = ∛(((400 * 24 * 60 * 60)² * (6.67430 x 10^-11)) / (4π² * (6.00 * 10^30 + 8.00 * 10^22)))
= 2.46 x 10^11 metres
Therefore, the orbital radius of the planet is approximately 2.46 x 10^11 meters.
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Calculate the force between 2 charges which each have a charge of +2.504C and
are separated by 1.25cm.
The force between the two charges of +2.504 C, separated by 1.25 cm, is approximately [tex]3.0064 \times 10^{14}[/tex] Newtons.
To calculate the force between two charges, we can use Coulomb's law, which states that the force (F) between two charges (q1 and q2) is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The formula for Coulomb's law is:
[tex]F = \frac {(k \times q_1 \times q_2)}{r^2}[/tex] where F is the force, k is the electrostatic constant (approximately [tex]9 \times 10^9 N \cdot m^2/C^2[/tex]), q₁ and q₂ are the charges, and r is the distance between the charges.
In this case, both charges have a value of +2.504 C, and they are separated by a distance of 1.25 cm (which is equivalent to 0.0125 m). Substituting these values into the formula, we have:
[tex]F = \frac{(9 \times 10^9 N \cdot m^2/C^2 \times 2.504 C \times 2.504 C)}{(0.0125 m)^2}[/tex]
Simplifying the calculation, we find: [tex]F \approx 3.0064 \times 10^{14}[/tex] Newtons.
So, to calculate the force between two charges, we can use Coulomb's law. By substituting the values of the charges and the distance into the formula, we can determine the force. In this case, the force between the two charges of +2.504 C, separated by 1.25 cm, is approximately [tex]3.0064 \times 10^{14}[/tex] Newtons.
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7) Two charges, a−Q at x=−a,y=0 and a+2Q at x=+a,y=0. A third charge +Q is placed at the origin. What is the direction of the net force on the charge at the origin? A) along +x axis B) along −x−axis C) no dircction as it is zero
The direction of the net force on the charge at the origin is along the -x axis. Therefore the correct option is B) along -x-axis.
According to Coulomb's Law, the electric force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. The direction of the force is along the line joining the two charges and acts from the charge with higher magnitude to the charge with lower magnitude.
In this scenario, the charge -Q at position (-a, 0) and the charge +2Q at position (+a, 0) exert forces on the charge +Q at the origin (0, 0). The force exerted by the charge -Q is attractive, directed towards the origin, while the force exerted by the charge +2Q is repulsive, directed away from the origin.
Since the force from the charge -Q is greater in magnitude compared to the force from the charge +2Q (due to the distances involved), the net force on the charge at the origin will be in the direction of the force from the charge -Q, which is along the -x axis (Option B).
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A crown weighing 8.30 N is suspended underwater from a string. The tension in the string is measured to be 7.81 N. Calculate the density of the crown either in gm/cc or kg/m3.
To calculate the density of the crown, we can use the concept of buoyancy. When an object is submerged in a fluid, the buoyant force exerted on the object is equal to the weight of the fluid displaced by the object.
In this case, the tension in the string is equal to the buoyant force acting on the crown, and the weight of the crown is given. By applying the equation for density, density = mass/volume, we can determine the density of the crown.
The buoyant force acting on the crown is equal to the tension in the string, which is measured to be 7.81 N. The weight of the crown is given as 8.30 N. According to Archimedes' principle, the buoyant force is equal to the weight of the fluid displaced by the crown. Therefore, the buoyant force can be considered as the difference between the weight of the fluid displaced and the weight of the crown.
The weight of the fluid displaced by the crown is equal to the weight of the crown when it is fully submerged. Thus, the weight of the fluid displaced is 8.30 N. Since the buoyant force is equal to the weight of the fluid displaced, it is also 8.30 N.
The density of an object is given by the equation density = mass/volume. In this case, the mass of the crown can be calculated using the weight of the crown and the acceleration due to gravity. The mass is given by mass = weight/gravity, where gravity is approximately 9.8 m/s^2. Therefore, the mass of the crown is 8.30 N / 9.8 m/s^2.
Finally, we can calculate the density of the crown by dividing the mass of the crown by its volume. The volume of the crown is equal to the volume of the fluid displaced, which is given by the formula volume = weight of the fluid displaced / density of the fluid. The density of water is approximately 1000 kg/m^3.
By substituting the values into the equation density = mass/volume, we can determine the density of the crown in either gm/cc or kg/m^3.
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A proton is released from rest between two charged plates where
the electric field has a strength of 300 N/C. When the proton moves
1.5 cm toward the negative plate, what is its speed?
The speed of the proton, when it moves 1.5 cm toward the negative plate, is approximately 2.25 x 10^7 m/s.
The speed of the proton can be determined using the principles of electrostatics and motion under constant acceleration.
Electric field strength (E) = 300 N/C
Distance moved by the proton (d) = 1.5 cm = 0.015 m (since it moves towards the negative plate, it moves opposite to the electric field)
Initial velocity (u) = 0 m/s (released from rest)
We can calculate the acceleration experienced by the proton using the equation:
Acceleration (a) = E / m
Where:
m is the mass of the proton (approximately 1.67 x 10^-27 kg)
Substituting the given values:
a = 300 N/C / (1.67 x 10^-27 kg)
Now, we can use the equations of motion to find the final velocity (v) of the proton.
v² = u² + 2ad
Since the proton starts from rest (u = 0), the equation simplifies to:
v² = 2ad
Substituting the known values:
v² = 2 * a * d
Calculating the values:
a = 300 N/C / (1.67 x 10^-27 kg)
v² = 2 * (300 N/C / (1.67 x 10^-27 kg)) * 0.015 m
v ≈ 2.25 x 10^7 m/s
Therefore, the speed of the proton, when it moves 1.5 cm toward the negative plate, is approximately 2.25 x 10^7 m/s.
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Points A and B lie between two infinite, uniformly charged
planes with surface charge densities ±σ. The potencial difference
ΔV = ΔA - ΔB is:
The potencial difference ΔV = ΔA - ΔB is:
ΔV = (σ/ε₀)•d
The expression for the potential difference between two points is given by ΔV= -∫E•dl where E is the electric field strength and dl is the infinitesimal displacement vector that leads from one point to the other point. This expression provides a clear indication that the potential difference is a path-dependent quantity, which means that the final result will vary depending on the path followed by dl. The potential difference between points A and B in the above-given figure can be calculated using the following expression: ΔV = -∫E•dl
Since the plates are uniformly charged, the electric field strength is constant in the region between the plates, and it points from the positive surface to the negative surface. We know that the electric field strength due to a uniformly charged plate is E=σ/2ε₀ where σ is the surface charge density of the plate and ε₀ is the electric permittivity of the free space. Thus, the electric field strength between the plates is given by E=σ/ε₀.
Since the path of dl lies perpendicular to the electric field strength E, we can simplify the above expression as follows: ΔV = -E•d where d is the distance between points A and B. Since the direction of the electric field strength is opposite to the direction of dl, we can simplify the above expression as follows: ΔV = E•dΔV = (σ/ε₀)•d The electric field strength between the plates is the same throughout the region between the plates.
Therefore, the potential difference between points A and B is given by ΔV = (σ/ε₀)•d.The potential difference between points A and B is ΔV = (σ/ε₀)•d.
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1. We will consider humanities ability to collect power from the Sun in this problem. The Sun has a luminosity of L = 3.846 x 1028 W, and a diameter of 1.393 million km. (a) Using the inverse-square law for intensities, , what is the intensity of sunlight when it reaches Earth at a distance of 149 million km from the Sun? Give your answer in W. (b) Now consider that the average total annual U.S. energy consumption is 2.22 x 1021 ). So, what is the average power requirement for the United States, in watts? (c) If solar cells can convert sunlight into electrical power at 30.0% efficiency, then how much total land area would need to be covered in solar cells to entirely meet the United States power requirements? Give your answer in square km. (d) If, in the future, an array of solar cells with a total surface area of 50,000 km2 was positioned in orbit around the Sun at a distance of 10 million km, and this array converts sunlight into electricity at 60.% efficiency, then how much energy a year would this array generate? Give your answer in Joules.
The answer is joules/year≈ 2.60 × 10²⁰J
(a) Using the inverse-square law for intensities, the intensity of sunlight when it reaches Earth at a distance of 149 million km from the Sun is given by the formula
I = L/(4πd²).
Here, L = 3.846 × 10²⁸ W, and
d = 149 × 10⁶ km
= 1.49 × 10⁸ km.
Plugging these values into the formula we get;
I = L/(4πd²)
= (3.846 × 10²⁸)/(4 × π × (1.49 × 10⁸)²)
≈ 1.37 kW/m²
(b) The average total annual U.S. energy consumption is 2.22 × 10²¹.
To get the average power requirement, we divide the energy consumption by the number of seconds in a year.
Thus, the average power requirement for the United States is given by:
P = (2.22 × 10²¹ J/year)/(365 × 24 × 60 × 60 seconds/year)
≈ 7.03 × 10¹¹ W
(c) If solar cells can convert sunlight into electrical power at 30.0% efficiency, then the amount of electrical power that can be generated per unit area of the solar cell is 0.3 kW/m².
To find the total land area needed to generate the entire US power requirements, we divide the power requirement by the power per unit area.
Thus, the total land area that would need to be covered in solar cells to entirely meet the United States power requirements is given by;
Area = (7.03 × 10¹¹ W)/(0.3 kW/m²)
≈ 2.34 × 10¹⁵ m²
= 2.34 × 10³ km²
(d) An array of solar cells with a total surface area of 50,000 km² was positioned in orbit around the Sun at a distance of 10 million km and converts sunlight into electricity at 60.% efficiency.
To calculate the total energy generated, we multiply the power generated by the area of the array and the number of seconds in a year.
Hence, the energy generated by the array is given by;
Energy = Power × Area × (365 × 24 × 60 × 60 seconds/year)
where Power = (0.6 × 1.37 kW/m²)
= 0.822 kW/m²
Area = 50,000 km² = 50 × 10⁶ m²
Therefore; Energy = 0.822 × 50 × 10⁶ × (365 × 24 × 60 × 60) Joules/year
≈ 2.60 × 10²⁰J
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C.2 a) A particular optical device has Jones matrix J = when expressed using a standard Cartesian co-ordinate system. i) Find the polarization state transmitted by this device when light linearly po- larized along the x direction is incident upon it. [2] ii) Repeat for y polarized incident light. iii) Find the eigenpolarizations and eigenvalues of J. [5] iv) On the basis of your results for parts i)-iii), identify the device, and suggest the physical effect responsible for its behaviour. [3] v) An unpolarised light beam of intensity Io passes through our device and is near-normally incident upon a high quality mirror, as illustrated below. Mirror Unpolarised Given that the Jones matrix of the device is the same for travel in either direction, express I₁ (the intensity after the first pass) and I2 (the intensity after the second pass) in terms of the incident intensity Io. [4] b) The E field of a particular electromagnetic wave has the form: E(z,t) = [e, cos(wt) + e, sin(wt)] Eo sin(kz) (i) Sketch the t dependence of E, vs E, for a series of different values of z. (ii) Sketch also Ez(z) vs Ey(2) for a series of different values of t. [2] [2] [2]
(a)i) When x polarized light is incident on the device, the transmitted light polarization state is given by T
=Jx
= [1 0; 0 1/3] [1; 0]
= [1; 0].ii) When y polarized light is incident on the device, the transmitted light polarization state is given by T
=Jy
= [1/3 0; 0 1] [0; 1]
= [0; 1]
=0,
= 0; Therefore, the eigenvalues of J are λ₁
=1 and λ₂
=1/3. Corresponding to these eigenvalues, we find the eigenvectors by solving (J-λ₁I) p₁
=0 and (J-λ₂I) p₂
=0. Thus, we get: p₁
= [1; 0] and p₂
=[0; 1]. iv) The device is a polarizer with polarization directions along x and y axes. T
= |T|²Io
= 1/3 Io. The reflected beam is also unpolarized, so its intensity is also 1/3 Io.
= 2/3 Io.
=λ/4, E z has maximum amplitude and is in phase with Ey, while at z
=3λ/4, Ez has minimum amplitude and is out of phase with Ey.
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Location A is 3.00 m to the right of a point charge q. Location B lies on the same line and is 4.00 m to the right of the charge. The potential difference between the two locations is VB - VA = 45 V. Determine q.
We can use the formula to determine the potential difference between two points due to an electric field caused by a point charge,q. The value of q is 5 × 10^-8 C.
The formula is:
[tex]V = kq/r[/tex],
where V is the potential difference, k is Coulomb's constant, q is the charge, and r is the distance between the two points.
The potential difference between location A and location B is given as VB - VA = 45 V.
Let's assume that the distance between the point charge and location A is x meters.
So, the distance between the point charge and location B would be (x + 4) meters.
Using the formula, the potential difference between the two points can be written as:
[tex]VB - VA = V(x + 4) - V(x)[/tex]
= V(4)
= kq(4 + x)/x
Let's assume that the value of k is 9 × 10^9 Nm^2/C^2.
Substituting the values, we get: 45 = (9 × 10^9 × q × (x + 4))/x
Solving this equation for q, we get: q = 5 × 10^-8 C.
So, the value of q is 5 × 10^-8 C.
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In describing his upcoming trip to the Moon, and as portrayed in the movie Apollo 13 (Universal, 1995 ), astronaut Jim Lovell said, "I'll be walking in a place where there's a 400 -degree difference between sunlight and shadow." Suppose an astronaut standing on the Moon holds a thermometer in his gloved hand.(b) Does it read any temperature? If so, what object or substance has that temperature?
According to astronaut Jim Lovell, "I'll be walking in a place where there's a 400-degree difference between sunlight and shadow.
Suppose an astronaut standing on the Moon holds a thermometer in his gloved hand. If so, what object or substance has that temperature?Astronauts on the Moon's surface will encounter extreme temperatures ranging from approximately .
However, the spacesuit has a cooling and heating system, as well as insulation materials that prevent the body from overheating or cooling too rapidly in the vacuum of space.Therefore, the thermometer in an astronaut's gloved hand would most likely read the temperature of the spacesuit material and not the extreme temperatures on the lunar surface.
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A car, initially at rest, accelerates at 3.34 m/s2 for 12 1 s How far did in go in this time?
The car traveled a distance of 23.96 meters in this time.
To determine the distance traveled by the car, we can use the formula of motion for constant acceleration: d = v0 * t + (1/2) * a * t^2, where d is the distance traveled, v0 is the initial velocity (which is zero in this case), t is the time, and a is the acceleration.
Plugging in the values, we have: d = 0 * 12.1 s + (1/2) * 3.34 m/s^2 * (12.1 s)^2.
Simplifying the equation, we get: d = (1/2) * 3.34 m/s^2 * (146.41 s^2) = 244.4947 m.
Rounding to two decimal places, the distance traveled by the car in this time is approximately 23.96 meters.
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Object A, which has been charged to +13.96 nC, is at the origin.
Object B, which has been charged to -25.35 nC, is at x=0 and y=1.42
cm. What is the magnitude of the electric force on object A?
the magnitude of the electric force on Object A is 0.0426 N.
Given data:Object A charge = +13.96 nC.Object B charge = -25.35 nC.Object B location = (0, 1.42) cm.The formula used to find the magnitude of the electric force is:
F = k * q1 * q2 / r^2 where k is Coulomb's constant which is equal to 9 x 10^9 Nm^2/C^2.q1 and q2 are the charges of object A and object B, respectively.r is the distance between the objects.
To find the distance between Object A and Object B, we use the distance formula which is:d = sqrt((x2 - x1)^2 + (y2 - y1)^2)where x1 and y1 are the coordinates of Object A (which is at the origin) and x2 and y2 are the coordinates of Object B.Using the given data, we can calculate:d = sqrt((0 - 0)^2 + (1.42 - 0)^2)d = 1.42 cm = 0.0142 m
Now we can substitute all the values into the formula:F = k * q1 * q2 / r^2F = (9 x 10^9 Nm^2/C^2) * (13.96 x 10^-9 C) * (-25.35 x 10^-9 C) / (0.0142 m)^2F = -4.26 x 10^-2 N = 0.0426 N (to three significant figures)
Therefore, the magnitude of the electric force on Object A is 0.0426 N.
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The magnitude of the electric force on object A is 8.10×10⁻² N.
The electric force between two charges can be determined using Coulomb's Law which is defined as F = k q1 q2 / r², where F is the force exerted by two charges, q1 and q2, k is the Coulomb constant, and r is the distance between the two charges.
Coulomb's Law states that the electric force between two charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
The electric force between object A and object B is given as F = k(q1q2 / r²)
Here, q1 = 13.96 nC and q2 = -25.35 nC.
Therefore, the electric force between object A and object B is given as F = k q1 q2 / r²
F = 9 x 10⁹ (13.96 x 10⁻⁹) (25.35 x 10⁻⁹) / (0.0142)²
F = 8.10 x 10⁻² N.
Thus, the magnitude of the electric force on object A is 8.10×10⁻² N.
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suppose a 42.5 cm long, 9.5 cm diameter solenoid has 1000 loops. how fast can it be turned off (in s) if the average induced emf cannot exceed 2.8v? assume there is an inital current of 21.5 A passing through the solenoid.
Given data, Length of solenoid l = 42.5 cm Diameter of solenoid d = 9.5 cm Radius of solenoid r = d/2 = 4.75 cm Number of turns n = 1000Current i = 21.5 A Induced EMF e = 2.8 V .
Here, L is the inductance of the solenoid .We know that the inductance of a solenoid is given by[tex]L = (μ0*n^2*A)[/tex]/where, μ0 is the permeability of free space n is the number of turns per unit length A is the cross-sectional area of the solenoid is the length of the solenoid Hence,
H Now, let's calculate the rate of change of[tex]current using e = -L(di/dt)di/dt = -e/L = -2.8/6.80= -0.4118[/tex]A/s Using [tex]i = i0 + (di/dt) × t i = 21.5 A, i0 = 0, and di/dt = -0.4118 A/st= i0/(di/dt) = 0 / (-0.4118)= 0 s[/tex] Therefore, the solenoid cannot be turned off as the average induced EMF cannot exceed 2.8 V.
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The resolving power of a refracting telescope increases with the diameter of the spherical objective lens. In reality, it is impractical to increase the diameter of the objective lens beyond approximately 1 m. Why?
a. If the objective lens is too large, it is difficult to keep it clean.
b. The resulting increase in light scattering from the surface of the objective lens will blur the image.
c. The spherical objective lens should be replaced by a paraboloidal objective lens beyond a 1-m diameter.
d. The increasing size of the objective lens will cause chromatic aberration to grow worse than spherical aberration.
e. The resultant sagging of the mirror will cause spherical aberration.
The diameter of the spherical objective lens in a refracting telescope is limited to approximately 1 m due to the resulting increase in light scattering from the lens surface, which blurs the image.
Increasing the diameter of the objective lens beyond approximately 1 m leads to an increase in light scattering from the surface of the lens. This scattering phenomenon, known as diffraction, causes the light rays to deviate from their intended path, resulting in a blurring of the image formed by the telescope.
This limits the resolving power of the telescope, which is the ability to distinguish fine details in an observed object.
To overcome this limitation, alternative designs, such as using a paraboloidal objective lens instead of a spherical lens, can be employed. Paraboloidal lenses help minimize spherical aberration, which is the blurring effect caused by the lens not focusing all incoming light rays to a single point.
Therefore, the practical limitation of approximately 1 m diameter for the objective lens in refracting telescopes is primarily due to the increase in light scattering and the resulting image blurring.
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Fifteen identical particles have various speeds. One has a speed of 4.00 m/s, two have a speed of 5.00 m/s, three have a speed of 7.00 m/s, four have a speed of 5.00 m/s, three have a speed of 10.0 m/s and two have a speed of 14.0 m/s. Find (a) the average speed, (b) the rms speed, and (c) the most probable speed of these particles. (a) 7.50 m/s; (b) 8.28 m/s; (c) 14.0 m/s (a) 7.50 m/s; (b) 8.28 m/s; (c) 5.00 m/s (a) 7.53 m/s; (b) 8.19 m/s; (c) 14.0 m/s (a) 7.53 m/s; (b) 8.19 m/s; (c) 5.00 m/s Page 24 of 33
The correct answers are (a) 7.53 m/s, (b) 8.19 m/s, and (c) 5.00 m/s. The average speed is calculated as follows: v_avg = sum_i v_i / N
where v_avg is the average speed
v_i is the speed of particle i
N is the number of particles
Plugging in the given values, we get
v_avg = (4.00 m/s + 2 * 5.00 m/s + 3 * 7.00 m/s + 4 * 5.00 m/s + 3 * 10.0 m/s + 2 * 14.0 m/s) / 15
= 7.53 m/s
The rms speed is calculated as follows:
v_rms = sqrt(sum_i (v_i)^2 / N)
Plugging in the given values, we get
v_rms = sqrt((4.00 m/s)^2 + 2 * (5.00 m/s)^2 + 3 * (7.00 m/s)^2 + 4 * (5.00 m/s)^2 + 3 * (10.0 m/s)^2 + 2 * (14.0 m/s)^2) / 15
= 8.19 m/s
The most probable speed is the speed at which the maximum number of particles are found. In this case, the most probable speed is 5.00 m/s.
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An air bubble at the bottom of a lake 41,5 m doep has a volume of 1.00 cm the temperature at the bottom is 25 and at the top 225°C what is the radius of the bubble ist before it reaches the surface? Express your answer to two significant figures and include the appropriate units.
The radius of the bubble before it reaches the surface is approximately 5.4 × 10^(-4) m
The ideal gas law and the hydrostatic pressure equation.
Temperature at the bottom (T₁) = 25°C = 25 + 273.15 = 298.15 K
Temperature at the top (T₂) = 225°C = 225 + 273.15 = 498.15 K
Using the ideal gas law equation: PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin.
(P₁ * V₁) / T₁ = (P₂ * V₂) / T₂
P₁ = pressure at the bottom of the lake
P₂ = pressure at the surface (atmospheric pressure)
V₁ = volume of the bubble at the bottom = 1.00 cm³ = 1.00 × 10^(-6) m³
V₂ = volume of the bubble at the surface (unknown)
T₁ = temperature at the bottom = 298.15 K
T₂ = temperature at the top = 498.15 K
V₂ = (P₂ * V₁ * T₂) / (P₁ * T₁)
P₁ = ρ * g * h
P₂ = atmospheric pressure
ρ = density of water = 1000 kg/m³
g = acceleration due to gravity = 9.8 m/s²
h = height = 41.5 m
P₁ = 1000 kg/m³ * 9.8 m/s² * 41.5 m
P₂ = atmospheric pressure (varies, but we can assume it to be around 1 atmosphere = 101325 Pa)
V₂ = (P₂ * V₁ * T₂) / (P₁ * T₁)
V₂ = (101325 Pa * 1.00 × 10^(-6) m³ * 498.15 K) / (1000 kg/m³ * 9.8 m/s² * 41.5 m * 298.15 K)
V₂ ≈ 1.10 × 10^(-6) m³
The volume of a spherical bubble can be calculated using the formula:
V = (4/3) * π * r³
The radius of the bubble before it reaches the surface is approximately 5.4 × 10^(-4) m
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A cylinder of radius 10 cm has a thread wrapped around its edge. If the cylinder is initially at rest and begins to rotate with an angular acceleration of 1 rad/s2, determine the length of thread that unwinds in 10 seconds.
Given
,Radius of cylinder
= r = 10 cm = 0.1 mAngular acceleration of cylinder = α = 1 rad/s²Time = t = 10s
Let’s find the angle covered by the cylinder in 10 seconds using the formula:θ = ωit + 1/2 αt²whereωi = initial angular velocity = 0 rad/st = time = 10 sα = angular acceleration = 1 rad/s²θ = 0 + 1/2 × 1 × (10)² = 50 rad
Now, let's find the length of the
thread
that unwinds using the formula:L = θrL = 50 × 0.1 = 5 mTherefore, the length of the thread that unwinds in 10 seconds is 5 meters.
Here, we used the formula for the arc
length of a circle
, which states that the length of an arc (in this case, the thread) is equal to the angle it subtends (in radians) times the radius.
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A small sphere of charge q = +68 MC and mass m = 5.8 g is attached to a light string and placed in a uniform electric field E that makes an angle 0 = 37° with the horizontal. The opposite end of the string is attached to a wall and the sphere is in static equilibrium when the string is horizontal as in Fig-
ure P15.22. (a) Construct a free body diagram for the sphere. Find (b) the magnitude of the clectric field and (c) the ten-
sion in the string.
The electric field is 8.53 × 10^-13 N/C, and the tension in the string is 2.68 mN.
(a) Free body diagram of the sphere is shown below.
(b)The electric force on the sphere is given by: F_el=qE[downward direction]
And, The gravitational force on the sphere is given by: F_gravity=mg[upward direction]
At equilibrium, the net force on the sphere is zero.
Therefore, F_el=F_gravityq
E=mg
=>E=mg/q
=5.8×10^-3/(68×10^6)C
=8.53×10^-13NC-1
(c)The tension in the string is equal in magnitude to the net force on the sphere in the vertical direction.
Tension= F_vertical= F_gravity- F_el
Since the sphere is in equilibrium, the magnitude of the tension must be equal to the vertical component of the gravitational force.
Hence,
Tension= F_gravity
sinθ= mg
sinθ=5.8×10^-3×9.
81×sin37°=2.68×10^-3N
=2.68 mN
Therefore,The electric field is 8.53 × 10^-13 N/C, and the tension in the string is 2.68 mN.
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113 ft3/min water is to be delivered through a 250 foot long smooth pipe with a pressure drop of 5.2 psi. Determine the required pipe diameter as outlined using the following steps: a) Use 3 inches as your initial guess for the diameter of the pipe and indicate what your next guess would be. b) During design, it is determined that the actual pipeline will include 7 standard elbows and two open globe valves. Show how your calculations for part a) would need to be modified to account for these fittings.
a) The next guess for the pipe diameter would be Y inches.
b) The modified calculations would include the equivalent lengths of the fittings to determine the required pipe diameter.
To determine the required pipe diameter, we can use the Darcy-Weisbach equation, which relates the pressure drop in a pipe to various parameters including flow rate, pipe length, pipe diameter, and friction factor. We can iteratively solve for the pipe diameter using an initial guess and adjusting it until the calculated pressure drop matches the desired value.
a) Using 3 inches as the initial guess for the pipe diameter, we can calculate the friction factor and the resulting pressure drop. If the calculated pressure drop is greater than the desired value of 5.2 psi, we need to increase the pipe diameter. Conversely, if the calculated pressure drop is lower, we need to decrease the diameter.
b) When accounting for fittings such as elbows and valves, additional pressure losses occur due to flow disruptions. Each fitting has an associated equivalent length, which is a measure of the additional length of straight pipe that would cause an equivalent pressure drop. We need to consider these additional pressure losses in our calculations.
To modify the calculations for part a), we would add the equivalent lengths of the seven standard elbows and two open globe valves to the total length of the pipe. This modified length would be used in the Darcy-Weisbach equation to recalculate the required pipe diameter.
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How much work is needed to bring a + 5.0 µC point charge from infinity to a point 2.0 m away from a + 25 µC charge? (you may assume that it is moved at a constant, controlled velocity so that there is no change in kinetic energy)
The work required to bring a +5.0 µC point charge from infinity to a point 2.0 m away from a +25 µC charge is 6.38 × 10^-5 joules.
To calculate the work, we can use the equation: Work = q1 * q2 / (4πε₀ * r), where q1 and q2 are the charges, ε₀ is the permittivity of free space, and r is the distance between the charges. Plugging in the given values, we get Work = (5.0 µC * 25 µC) / (4πε₀ * 2.0 m). Evaluating the expression, we find the work to be 6.38 × 10^-5 joules.Therefore, the work required to bring the +5.0 µC point charge from infinity to a point 2.0 m away from the +25 µC charge is 6.38 × 10^-5 joules.
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Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is 36.2 degrees. Dog A exerts a force of 11.1 N , and dog B exerts a force of 5.7 N . Find the magnitude of the resultant force. Express your answer in newtons.
The magnitude of the resultant force in newtons that is exerted by the two dogs pulling horizontally on ropes attached to a post is 12.6 N.
How to find the magnitude of the resultant force?The sum of the two vectors gives the resultant vector. The formula to find the resultant force, R is R = √(A² + B² + 2AB cosθ).
Where, A and B are the magnitudes of the two forces, and θ is the angle between them.
The magnitude of the resultant force is 12.6 N. Let's derive this answer.
Given;
The force exerted by Dog A, A = 11.1 N
The force exerted by Dog B, B = 5.7 N
The angle between the two ropes, θ = 36.2°
Now we can use the formula to find the resultant force, R = √(A² + B² + 2AB cosθ).
Substituting the given values,
R = √(11.1² + 5.7² + 2(11.1)(5.7) cos36.2°)
R = √(123.21 + 32.49 + 2(11.1)(5.7) × 0.809)
R = √(155.7)R = 12.6 N
Therefore, the magnitude of the resultant force is 12.6 N.
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When an atom undergoes beta+ decay (positron emission) or beta- decay (electron emission) the positron or electron must come from the nucleus. These particles do not normal reside in the nucleus and are actually formed inside the nucleus. An electron is formed when a _________ decays to become a(n) _________ and a(n) _________. This occurs as a result of the re-arrangement of the fundamental particles that neutrons and protons are made from. These particles are called ________.
During beta decay, an electron is formed when a neutron decays to become a proton and an electron. This process involves the rearrangement of fundamental particles called quarks.
Beta decay occurs when an atom undergoes either beta+ decay (positron emission) or beta- decay (electron emission). In beta- decay, a neutron in the nucleus decays to become a proton, and in the process, an electron is formed. The neutron is composed of three fundamental particles called quarks (two down quarks and one up quark), while the proton is composed of two up quarks and one down quark.
During the decay process, one of the down quarks in the neutron changes into an up quark, converting the neutron into a proton. Simultaneously, an electron is formed as a result of this rearrangement. The electron is emitted from the nucleus with high energy, carrying away the excess energy released during the decay.
The formation of an electron during beta- decay is a consequence of the re-arrangement of quarks within the neutron and proton. Quarks are elementary particles that make up protons, neutrons, and other subatomic particles. They have electric charges and different flavors (up, down, charm, strange, top, bottom). In beta- decay, the transformation of a neutron into a proton involves the conversion of one type of quark into another, accompanied by the emission of an electron.
During beta- decay, an electron is formed when a neutron decays to become a proton and an electron. This process involves the rearrangement of fundamental particles known as quarks.
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If 2 grams of matter could be entirely converted to energy, how
much would the energy produce cost at 25 centavos per kWh?
if 2 grams of matter could be entirely converted to energy, it would produce energy with a cost of 12.5 million pesos at 25 centavos per kWh.
How do we calculate?we will make use of the energy equation developed by Albert Einstein:
E = mc²
E= energy,
m = mass,
c = speed of light =[tex]3.0 * 10^8[/tex] m/s
E = (0.002 kg) * ([tex]3.0 * 10^8[/tex]m/s)²
E =[tex]1.8 * 10^1^4[/tex] joules
1 kWh = [tex]3.6 * 10^6[/tex] joules
Energy in kWh = ([tex]1.8 * 10^1^4[/tex] joules) / ([tex]3.6 * 10^6[/tex] joules/kWh)
Energy in kWh =[tex]5.0 * 10^7[/tex] kWh
The Cost is then found as = ([tex]5.0 * 10^7[/tex] kWh) * (0.25 pesos/kWh)
Cost = [tex]1.25 * 10^7[/tex]pesos
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