In the figure, a frictionless roller coaster car of mass m = 938 kg tops the first hill with speed v0 = 14.9 m/s at height h = 31.5 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?
To calculate the work done by the gravitational force on the roller coaster car, we can use the formula:
Work = Force * Distance * cos(θ)
Now, let's break down the problem step by step:
(a) To calculate the work done by the gravitational force from the starting point to point A, we need to find the vertical distance the car has traveled.
Since the roller coaster car is moving on a frictionless track, the work done by the gravitational force only depends on the change in height.
The change in height from the starting point to point A is h - 0 = 31.5 m.
The force of gravity acts in the opposite direction of the displacement, so the angle θ between the force and displacement is 180 degrees.
Using the formula: Work = m * g * h * cos(180)
Where:
m = mass of the roller coaster car = 938 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height = 31.5 m
Work(a) = 938 * 9.8 * 31.5 * cos(180)
Work(a) = -293785.4 J
Therefore, the work done by the gravitational force from the starting point to point A is -293785.4 Joules.
(b) To calculate the work done by the gravitational force from point A to point B, we need to find the vertical distance the car has traveled.
The change in height from point A to point B is 0 - 31.5 = -31.5 m.
Since the car is moving downwards, we need to consider the force of gravity acting in the same direction as the displacement.
The angle θ between the force and displacement is 0 degrees.
Using the formula: Work = m * g * h * cos(0)
Work(b) = 938 * 9.8 * (-31.5) * cos(0)
Work(b) = -293785.4 J
Therefore, the work done by the gravitational force from point A to point B is -293785.4 Joules.
(c) To calculate the work done by the gravitational force from point B to point C, we need to find the vertical distance the car has traveled.
The change in height from point B to point C is -31.5 - 0 = -31.5 m.
Since the car is moving downwards, we need to consider the force of gravity acting in the same direction as the displacement.
The angle θ between the force and displacement is 0 degrees.
Using the formula: Work = m * g * h * cos(0)
Work(c) = 938 * 9.8 * (-31.5) * cos(0)
Work(c) = -293785.4 J
Therefore, the work done by the gravitational force from point B to point C is -293785.4 Joules.
(d) To find the gravitational potential energy of the car-Earth system at point B, we can use the formula: Gravitational Potential Energy = m * g * h
Where:
m = mass of the roller coaster car = 938 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height = 0 m
Gravitational Potential Energy(d) = 938 * 9.8 * 0
Gravitational Potential Energy(d) = 0 J
Therefore, the gravitational potential energy of the car-Earth system at point B is 0 Joules.
(e) Similarly, to find the gravitational potential energy of the car-Earth system at point A, we can use the formula: Gravitational Potential Energy = m * g * h
Where:
m = mass of the roller coaster car = 938 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height = -31.5 m
Gravitational Potential Energy(e) = 938 * 9.8 * (-31.5)
Gravitational Potential Energy(e) = -282873 J
Therefore, the gravitational potential energy of the car-Earth system at point A is -282873 Joules.