A) Assume the mass of a pendulum is 3kg. Calculate the work done by the pendulum on the mass from 0 degrees to 180 degrees ( you will need to integrate with respects to d(theta ) instead of dx
given: the mass at the end of the string experiences a restoring force given by: F=-mgsin(theta) i
B) compare the work done in part A to the amount of potential energy given to the mass as it moves from 0 degrees to 180 degrees using mgh
To calculate the work done by the pendulum on the mass as it moves from 0 degrees to 180 degrees, we need to integrate the force over the displacement. In this case, the displacement is related to the angle (theta) through which the pendulum swings.
A) Calculation of work done by the pendulum:
Given:
Mass of the pendulum (m) = 3 kg
Restoring force (F) = -mgsin(theta)i
The work done (W) is defined as the integral of the dot product between the force and the displacement vector. In this case, the force is F = -mgsin(theta)i and the displacement vector is d(theta).
Therefore, we have:
W = ∫ F · d(theta)
We can rewrite the force F as F = -mgsin(theta)i = -mgd(theta)sin(theta)i, as d(theta)i = sin(theta)i.
Now, we can calculate the work using this expression:
W = ∫ (-mgd(theta)sin(theta)) · (sin(theta)i) = -mg ∫ sin^2(theta) d(theta)
To evaluate this integral, we can use the trigonometric identity: sin^2(theta) = (1 - cos(2theta))/2
So, W = -mg/2 ∫ (1 - cos(2theta)) d(theta)
Integrating, we get:
W = -mg/2 [theta - (sin(2theta))/2]
Evaluating the limits of integration from 0 to 180 degrees (or 0 to pi radians), we can substitute these values into the equation:
W = -mg/2 [(pi) - (sin(pi))/2 - (0) + (sin(0))/2]
W = -mg/2 [pi - 0 + 0]
W = -mg/2 * pi
Substituting the given mass (m = 3 kg) and acceleration due to gravity (g = 9.8 m/s^2), we can calculate the work done:
W = -(3 kg)(9.8 m/s^2)/2 * pi
W ≈ -45.15 J (rounded to two decimal places)
B) Comparison with potential energy:
The potential energy (PE) of the pendulum is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
In this case, as the pendulum swings from 0 degrees (the lowest point) to 180 degrees (the highest point), the height (h) changes from the lowest point to the highest point of the swing but the mass (m) remains constant.
Therefore, the work done by the pendulum on the mass (-45.15 J) corresponds to the decrease in potential energy of the system. Since the pendulum swings from a lower position to a higher position, the potential energy increases.
Hence, the work done by the pendulum (-45.15 J) is equal in magnitude but opposite in sign to the change in potential energy given to the mass as it moves from 0 degrees to 180 degrees.