A 2.20 kg pendulum starts from a height of 5.00m. It swings back and forth through one whole oscillation but only returns to a maximum height of 4.75m. After a long time the pendulum eventually winds down and comes to a stop. How much total negative work was done on the pendulum to bring it to rest at its lowest point?
I know that negative work during the first oscillation is 5.39 J.
It is the potential energy lost, m g h
Well, it seems like the pendulum needs to take some "rest" to gather its energy for the next performance. But let's not be negative about it! In order to find the total negative work done on the pendulum to bring it to rest at its lowest point, we need to consider the difference in potential energy between its starting height and the lowest point it reaches.
The potential energy at the highest point is given by mgh, where m is the mass (2.20 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height (5.00 m). So the potential energy at the highest point is 2.20 kg * 9.8 m/s² * 5.00 m = 107.8 J.
Now, at the lowest point, the potential energy is mgh, with m and g unchanged, but the height is 4.75 m. So the potential energy at the lowest point is 2.20 kg * 9.8 m/s² * 4.75 m = 103.57 J.
To find the negative work done, we take the difference between the initial and final potential energies: 107.8 J - 103.57 J = 4.23 J.
So, it looks like the pendulum experienced some "down time" with a total negative work of 4.23 J to bring it to rest at its lowest point. Keep swinging, pendulum!
To calculate the total negative work done on the pendulum, we need to consider the work done during the first oscillation and the work done as it comes to rest at its lowest point.
Given:
Mass of the pendulum, m = 2.20 kg
Height from which it starts, h1 = 5.00 m
Maximum height it returns to, h2 = 4.75 m
Let's calculate the work done during the first oscillation:
Work done during first oscillation = Mass x Acceleration due to gravity x Change in height
The change in height can be obtained by subtracting the final height (h2) from the initial height (h1).
Change in height = h1 - h2 = 5.00 m - 4.75 m = 0.25 m
Plugging in the values:
Work done during first oscillation = 2.20 kg x 9.81 m/s^2 x 0.25 m
= 5.39 J (as you mentioned)
Now, to determine the work done as the pendulum comes to rest at its lowest point, we need to find the change in potential energy from the initial height (h1) to the lowest point (zero height).
Change in potential energy = Mass x Acceleration due to gravity x Change in height
Since the pendulum comes to rest at the lowest point, the change in height is h1 - 0 = h1.
Plugging in the values:
Change in potential energy = 2.20 kg x 9.81 m/s^2 x 5.00 m
= 108.9 J
Now, since the change in potential energy is negative (going from a higher height to the lowest point), the total negative work done on the pendulum to bring it to rest at the lowest point is the sum of the work done during the first oscillation and the change in potential energy:
Total negative work done = Work done during first oscillation + Change in potential energy
= 5.39 J + (-108.9 J)
≈ -103.51 J
Thus, the total negative work done on the pendulum to bring it to rest at its lowest point is approximately -103.51 J.
To determine the total negative work done on the pendulum to bring it to rest at its lowest point, we need to calculate the total work done against gravity during the entire motion.
First, we calculate the change in potential energy of the pendulum from its highest point to its lowest point. The change in potential energy can be calculated using the equation: ΔU = mgh, where m is the mass (2.20 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the change in height (5.00 m - 4.75 m = 0.25 m).
ΔU = (2.20 kg)(9.8 m/s²)(0.25 m) = 5.39 J
This value is consistent with the negative work you stated during the first oscillation. However, to find the total negative work over multiple oscillations, we need to consider that the work done against gravity is equal for each full oscillation of the pendulum.
Thus, we can conclude that the total negative work done on the pendulum to bring it to rest at its lowest point is also 5.39 J.