In the figure, a frictionless roller coaster car of mass m = 672 kg tops the first hill with speed v0 = 19.4 m/s at height h = 34.7 m. How much work does the gravitational force do on the car from that point to (a) point A, (b) point B, and (c) point C? If the gravitational potential energy of the car-Earth system is taken to be zero at C, what is its value when the car is at (d) B and (e) A?
(a) W(grav) = 0 since there is no change in height between the initial point and point A.
(b) W(grav) – mgh –mg(h/2) = mgh/2
(c) W(grav) mgh -0 = mgh
(d) mgh/2
(e) mgh
To calculate the work done by the gravitational force on the roller coaster car, we can use the concept of work-energy theorem. The work done by a force is equal to the change in kinetic energy.
(a) Point A:
Since the roller coaster starts at rest at point A, the initial kinetic energy is zero. The work done by the gravitational force from the starting point to A is therefore equal to the change in potential energy:
Work A = ΔPE_A = mgh_A
(b) Point B:
To calculate the work done by the gravitational force from A to B, we need to consider both the change in potential energy and the change in kinetic energy. The work done is equal to the sum of these two changes:
Work B = ΔPE_B + ΔKE_B = mgh_B + (1/2)mv_B^2
(c) Point C:
Similar to point A, the roller coaster comes to rest at point C. Therefore, the work done by the gravitational force from B to C is equal to the negative change in potential energy:
Work C = -ΔPE_C = -mgh_C
(d) Potential energy at point B:
To find the potential energy at point B, we can use the equation for gravitational potential energy:
PE_B = mgh_B
(e) Potential energy at point A:
Similarly, to find the potential energy at point A, we can use the equation for gravitational potential energy:
PE_A = mgh_A
Now, let's plug in the given values:
m = 672 kg
v0 = 19.4 m/s
h = 34.7 m
(a) Work A = mgh_A = 672 kg * 9.8 m/s^2 * 34.7 m
(b) Work B = mgh_B + (1/2)mv_B^2 = 672 kg * 9.8 m/s^2 * h_B + (1/2) * 672 kg * v_B^2
(c) Work C = -mgh_C = -672 kg * 9.8 m/s^2 * h_C
(d) PE_B = mgh_B = 672 kg * 9.8 m/s^2 * h_B
(e) PE_A = mgh_A = 672 kg * 9.8 m/s^2 * h_A
Just replace h_A, h_B, and h_C with their respective heights in meters to find the numerical values.
To determine the amount of work done by the gravitational force on the roller coaster car at various points in the system, we can use the principles of work and energy.
(a) Work done on the car from the top of the hill to Point A:
The work done by the gravitational force can be calculated using the work-energy principle, which states that the work done on an object is equal to the change in its mechanical energy. In this case, the mechanical energy consists of both kinetic energy (KE) and gravitational potential energy (PE).
The total mechanical energy at the top of the hill can be written as the sum of the initial kinetic energy and potential energy:
E_initial = KE_top + PE_top
Since the roller coaster car is at the top of the hill, its velocity is zero, and hence its kinetic energy is also zero:
KE_top = 0
The gravitational potential energy at the top of the hill can be calculated using the formula:
PE_top = m * g * h
Where m is the mass of the car, g is the acceleration due to gravity, and h is the height.
Plugging in the values given:
PE_top = 672 kg * 9.8 m/s^2 * 34.7 m
The work done from the top of the hill to Point A is the change in mechanical energy:
Work_A = E_initial - E_A
Since the roller coaster car is initially at rest at the top, its initial mechanical energy is equal to its potential energy:
E_initial = PE_top
The mechanical energy at Point A is the sum of the kinetic energy and potential energy:
E_A = KE_A + PE_A
At Point A, the car will have both kinetic and potential energy. To calculate these values, we can use the conservation of mechanical energy, assuming no frictional forces. The total mechanical energy at Point A is equal to the sum of the kinetic and potential energy at the top of the hill:
E_A = E_initial (conservation of mechanical energy)
Therefore, the work done by the gravitational force from the top of the hill to Point A is:
Work_A = PE_top - E_A
(b) Work done on the car from the top of the hill to Point B:
Using the same process as above, we can calculate the work done by the gravitational force from the top of the hill to Point B.
The mechanical energy at Point B is equal to the sum of the initial kinetic energy and the potential energy:
E_B = KE_B + PE_B
By using the conservation of mechanical energy:
E_B = E_initial
Therefore, the work done by the gravitational force from the top of the hill to Point B is:
Work_B = PE_top - E_B
(c) Work done on the car from the top of the hill to Point C:
At Point C, the car's gravitational potential energy is taken to be zero.
Since potential energy at Point C is zero, the work done by the gravitational force from the top of the hill to Point C is:
Work_C = PE_top
(d) Gravitational potential energy of the car at Point B:
The gravitational potential energy at Point B can be calculated using the same formula as before:
PE_B = m * g * h
(e) Gravitational potential energy of the car at Point A:
The gravitational potential energy at Point A can also be calculated using the formula:
PE_A = m * g * h
Note that in both cases, the height (h) is measured relative to the height of Point C, where the gravitational potential energy is taken to be zero.
By plugging in the values and performing the calculations using the given information, you can find the specific numerical answers for (a), (b), (c), (d), and (e).