Question

A pendulum is made up of a small sphere of mass 0.500 kg attached to a string of length 0.950 m. The sphere is swinging back and forth between point A, where the string is at the maximum angle of 35.0∘ to the left of vertical, and point C, where the string is at the maximum angle of 35.0∘ to the right of vertical. The string is vertical when the sphere is at point B. Calculate how much work the force of gravity does on the sphere from A to B.

Answer 1

The work done by gravity on the sphere from point A to point B in a pendulum swing is the change in gravitational potential energy, which can be calculated using the mass of the sphere, the acceleration due to gravity, and the change in height based on the pendulum's length and the angle from the vertical. Using the formulas ΔPE = mgh and h = L(1 - cos(θ)), the work done is approximately 1.71 Joules.

Step-by-step explanation:

To calculate the work done by gravity on the sphere as it swings from point A to point B on a pendulum, we assume that gravitational force is conservative, which means work done is independent of the path taken. Work done by gravity is equal to the change in potential energy. As the sphere moves from point A to point B, the height decreases from its maximum value at A to the lowest value at B, so the potential energy decreases. To calculate the change in potential energy, we use the formula:

ΔPE = mgh

where:

• m is the mass of the sphere
• g is the acceleration due to gravity (9.81 m/s^2)
• h is the change in height

The height change can be found using trigonometry by calculating the vertical component of the initial position of the pendulum. Assuming small angles for simplicity, where the sine of the angle is approximately equal to the angle (in radians), we have:

h = L(1 - cos(θ))

where:

• L is the length of the pendulum
• θ is the angle in degrees, converted to radians

The work done by gravity from A to B, which is the same as the change in potential energy, would be:

W = ΔPE = mgh

Rearranging and substituting in the known values (assuming θ is small for simplicity), we get:

W = 0.500 kg * 9.81 m/s^2 * 0.950 m * (1 - cos(35.0°))

Converting 35 degrees to radians and performing the calculation gives:

W = 0.500 kg * 9.81 m/s^2 * 0.950 m * (1 - cos(0.610865 rad)) ≈ 1.71 J

The work done by the force of gravity on the sphere from A to B is approximately 1.71 Joules.

Answer 2

W = 0.842 J

Step-by-step explanation:

To solve this exercise we can use the relationship between work and kinetic energy

W = ΔK

In this case the kinetic energy at point A is zero since the system is stopped

W = K_f (1)

now let's use conservation of energy

starting point. Highest point A

Em₀ = U = m g h

Final point. Lowest point B

Em_f = K = ½ m v²

energy is conserved

Em₀ = Em_f

mg h = K

to find the height let's use trigonometry

at point A

cos 35 = x / L

x = L cos 35

so at the height is

h = L - L cos 35

h = L (1-cos 35)

we substitute

K = m g L (1 -cos 35)

we substitute in equation 1

W = m g L (1 -cos 35)

let's calculate

W = 0.500 9.8 0.950 (1 - cos 35)

W = 0.842 J