Figure 8-31 shows a ball with mass m=0.341 kg attached to the end of a thin rod with length L=0

Figure 8-31 shows a ball with mass m=0.341 kg attached to...

Figure $8-31$ shows a ball with mass $m=0.341 \mathrm{~kg}$ attached to the end of a thin rod with length $L=0.452 \mathrm{~m}$ and negligible mass. The other end of the rod is pivoted so that the ball can move in a vertical circle. The rod is held horizontally as shown and then given enough of a downward push to cause the ball to swing down and around and just reach the vertically up position, with zero speed there. How much work is done on the ball by the gravitational force from the initial point to (a) the lowest point, (b) the highest point, and (c) the point on the right level with the initial point? If the gravitational potential energy of the ball-Earth system is taken to be zero at the initial point, what is it when the ball reaches (d) the lowest point, (e) the highest point, and (f) the point on the right level with the initial point? (g) Suppose the rod were pushed harder so that the ball passed through the highest point with a nonzero speed. Would $\Delta U_{g}$ from the lowest point to the highest point then be greater than, less than, or the same as it was when the ball stopped at the highest point?

Figure $8-31$ shows a ball with mass $m=0.341 \mathrm{~kg}$ attached to the end of a thin rod with length $L=0.452 \mathrm{~m}$ and negligible mass. The other end of the rod is pivoted so that the ball can move in a vertical circle. The rod is
held horizontally as shown and then given enough of a downward push to cause the ball to swing down and around and just reach the vertically up position, with zero speed there. How much work is done on the ball by the gravitational force from the initial point to (a) the lowest point, (b) the highest point, and (c) the point on the right level with the initial point? If the gravitational potential energy of the ball-Earth system is taken to be zero at the initial point, what is it when the ball reaches (d) the lowest point, (e) the highest point, and (f) the point on the right level with the initial point? (g) Suppose the rod were pushed harder so that the ball passed through the highest point with a nonzero speed. Would $\Delta U_{g}$ from the lowest point to the highest point then be greater than, less than, or the same as it was when the ball stopped at the highest point?

Key Concepts

Work Done by Gravity

Work done by gravity is calculated as the product of the gravitational force and the vertical displacement of an object. Since gravity points downward, the work done by gravity depends solely on the change in vertical height, making it a path-independent (conservative) force. This concept is fundamental in analyzing energy changes in systems involving vertical motion.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field, typically given by the formula U = mgh, where h is the height relative to a chosen reference level. Changes in GPE reflect the work done against or by the gravitational force and are central to understanding energy exchanges in vertical movements.

Conservative Forces and Path Independence

Gravity is a conservative force, meaning that the work it does on an object depends only on the starting and ending positions, not on the path taken. This property is pivotal because it allows us to use potential energy functions to analyze energy changes, ensuring that the difference in gravitational potential energy depends only on vertical displacement.

Reference Levels in Potential Energy

The choice of a zero reference point for potential energy is arbitrary and does not affect the physical results, as only differences in potential energy are physically meaningful. Setting the potential energy to zero at a particular point simplifies the analysis of energy changes in a system, making it easier to calculate work done and energy differences between positions.

Transcript

00:01 Looking at problem number four from chapter eight and this problem deals with a ball let's draw that right now so we have a let's just draw you know a circle right now and then we'll trace in the rest so you have a ball and it is connected to a ride this is the top okay so we're gonna we're going to actually say that this is the starting position of the ball.

00:36 And you know that this is the starting position because i am putting it in black.

00:42 So black is starting position.

00:44 And this is length of the rod.

00:48 And the problem tells us that the mass of this ball is 0 .341 kilograms and that the length of the rod is 0 .452 meters.

01:04 And it wants to know how much work is done by the ball.

01:09 No, how much work is done by the gravitational force on the ball as it moves through this circle.

01:18 So let's, you know, let's do it.

01:22 So it's problem a says that the lowest point.

01:26 So we're assuming that this is the lowest point here.

01:29 And if this is the lowest point, then, well, you know how to do this.

01:35 Work is equal to m .g .h.

01:40 So this ball is going to move a length of l, and all you have to do is basically substitute the values.

01:50 So 0 .341 kilograms, gravitational constant, and then the length.

02:00 0 .452 meters, and that is literally it.

02:05 And now b asks for the highest point, which we know is right here.

02:11 So this is the highest point, keep that in mind, and this is the lowest point.

02:19 Now the highest point is literally the same thing, except we need to keep in mind that it will be a negative value.

02:29 Now, why is that? it's moving the same distance.

02:34 It's moving with the same ball.

02:38 It's moving on earth.

02:39 So the gravitational constant is the same.

02:41 But why is it that it's negative? well, let's keep in mind that if there's work being done on the ball to move it upwards, it's actually going against gravity, which is why it's negative.

02:55 Whereas there's positive work done when it's going.

02:59 To the bottom because it's easier.

03:02 It's going with gravity and that's why it's positive.

03:05 So you know that the answer to this will literally be the answer to a, but with a negative.

03:15 Now c asks for how much work is being done if the ball is moved from this point, the initial starting point.

03:25 So let's write initial and then.....

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