A 1200 wrecking ball hangs from a 20 -long cable. The ball is pulled back until the cable makes an angle of 30.0 with the vertical.By how much has the gravitational potential energy of the ball changed?

draw the figure. From the swing point, the initial point was 20 below, at the new point, it is 20cos30 below. What is the difference?

To calculate the change in gravitational potential energy, we need to find the difference between the initial and final gravitational potential energies.

The gravitational potential energy of an object near the Earth's surface is given by the equation:

PE = m * g * h

Where:
PE is the gravitational potential energy,
m is the mass of the object,
g is the acceleration due to gravity, and
h is the height.

Given:
Mass of the wrecking ball, m = 1200 kg
Length of the cable, h = 20 m
Angle made by the cable with the vertical, θ = 30°

First, let's find the height of the wrecking ball above the ground when it's at an angle of 30° with the vertical. We can use basic trigonometry:

h' = h * sin(θ)
h' = 20 * sin(30°)
h' ≈ 10 m

Now, let's calculate the initial gravitational potential energy:

PE_initial = m * g * h

PE_initial = 1200 kg * 9.8 m/s² * 20 m
PE_initial = 235,200 J

Next, let's calculate the final gravitational potential energy:

PE_final = m * g * h'

PE_final = 1200 kg * 9.8 m/s² * 10 m
PE_final = 117,600 J

Finally, calculate the change in gravitational potential energy:

ΔPE = PE_final - PE_initial
ΔPE = 117,600 J - 235,200 J
ΔPE = -117,600 J

The change in gravitational potential energy of the ball is -117,600 J. Since the value is negative, it means that the gravitational potential energy has decreased.

To determine how much the gravitational potential energy of the ball has changed, we need to know the initial and final heights of the ball.

The initial height of the ball can be calculated using the given information. Since the cable makes an angle of 30.0° with the vertical, we can use trigonometry to find the height. The triangle formed by the cable, the height, and the vertical is a right-angled triangle.

Using the sine function, we can find the initial height:

sin(30°) = height / 20
height = 20 * sin(30°)
height ≈ 10 meters

Now we need to find the final height of the ball. Since the cable is pulled back, the height of the ball will change.

The change in height can be determined using the same trigonometric principles. In this case, the angle between the vertical and the cable is 90° - 30° = 60°.

We can again use the sine function to find the change in height:

sin(60°) = change in height / 20
change in height = 20 * sin(60°)
change in height ≈ 17.32 meters

Finally, we can calculate the change in the gravitational potential energy of the ball using the formula:

Change in gravitational potential energy = mass * gravitational acceleration * change in height

Given that the mass of the ball is 1200 kg and the acceleration due to gravity is approximately 9.8 m/s²:

Change in gravitational potential energy = 1200 kg * 9.8 m/s² * 17.32 meters
Change in gravitational potential energy ≈ 206,825.6 Joules

Therefore, the gravitational potential energy of the ball has changed by approximately 206,825.6 Joules.