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.
the pendulum looses energy (work) on each swing because the bob returns to a lower height each time
when the bob stops (at rest, at the lowest point), the total lost energy (negative work) is equal to the difference in potential energy between the starting point and the lowest point ... m g h
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To find the total negative work done on the pendulum, we need to calculate the work done on the pendulum during each oscillation and sum them up.
The work done on an object is given by the equation:
Work = force × displacement × cos(angle)
In this case, the force acting on the pendulum is the force due to gravity, which is equal to the weight of the pendulum. The displacement is the vertical distance the pendulum moves from its initial height to the lowest point, which is 5.00 m - 4.75 m = 0.25 m.
Since the force of gravity and the displacement are in the same direction, the angle between them is 0 degrees, so the cos(angle) term is equal to 1.
The weight of the pendulum can be calculated using the equation:
Weight = mass × acceleration due to gravity
where the mass is 2.20 kg and the acceleration due to gravity is approximately 9.8 m/s^2.
Weight = 2.20 kg × 9.8 m/s^2 = 21.56 N
Now we can calculate the work during each oscillation:
Work = 21.56 N × 0.25 m × cos(0 degrees) = 21.56 N × 0.25 m × 1 = 5.39 J
As you've mentioned, the work done during the first oscillation is 5.39 J.
Since the pendulum eventually comes to a stop after multiple oscillations, the total negative work done on the pendulum can be found by summing up the work done during each oscillation.
Total negative work = 5.39 J + 5.39 J + ...
Since the number of oscillations is not specified, we cannot provide an exact value for the total negative work. However, we can infer that the total negative work done on the pendulum will be the sum of multiple 5.39 J values, which will increase as the number of oscillations increases.
To calculate the total negative work done on the pendulum, you need to consider two parts: the work done during the first oscillation and the work done as the pendulum comes to rest at its lowest point.
You mentioned that the negative work during the first oscillation is 5.39 J. This is correct, but it only accounts for the work done during the swinging motion.
To calculate the work done as the pendulum comes to rest, you need to determine the change in gravitational potential energy. The initial height of the pendulum is 5.00 m, and it comes to rest at a height of 0.00 m (the lowest point). The change in height is therefore 5.00 m.
The gravitational potential energy is given by the formula U = mgh, where U is the energy, m is the mass, g is the acceleration due to gravity, and h is the height.
Plugging in the values, we have:
U = (2.20 kg) * (9.8 m/s^2) * (5.00 m)
U = 107.8 J
Since the pendulum comes to rest at the lowest point, all the potential energy is converted into negative work. Therefore, the total negative work done on the pendulum to bring it to rest at its lowest point is 107.8 J.
So, the total negative work done on the pendulum is 107.8 J + 5.39 J = 113.19 J.