In Fig. 8-33, a small block of mass m=0.032 kg can slide along the frictionless loop-the-loop,

In Fig. 8-33, a small block of mass m=0.032 kg can slide...

In Fig. 8-33, a small block of mass $m=0.032 \mathrm{~kg}$ can slide along the frictionless loop-the-loop, with loop radius $R=12 \mathrm{~cm}$. The block is released from rest at point $P$, at height $h=5.0 R$ above the bottom of the loop. How much work does the gravitational force do on the block as the block travels from point $P$ to (a) point $Q$ and (b) the top of the loop? If the gravitational potential energy of the block-Earth system is taken to be zero at the bottom of the loop, what is that potential energy when the block is (c) at point $P,(\mathrm{~d})$ at point $Q$, and $(\mathrm{e})$ at the top of the loop? (f) If, instead of merely being released, the block is given some initial speed downward along the track, do the answers to (a) through (e) increase, decrease, or remain the same?

Key Concepts

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy an object has due to its position in a gravitational field. It is typically given by the equation U = mgh, where h is the height above a chosen reference level. GPE is crucial in analyzing energy exchanges in mechanical systems and understanding how potential energy converts to kinetic energy as an object moves.

Work Done by Gravity

Work performed by gravity involves the force of gravity acting over a displacement. Because gravity is a conservative force, the work done depends only on the vertical displacement between the initial and final positions and equals the negative change in gravitational potential energy. This means that regardless of the path taken, the work done by gravity can be directly determined by the difference in height.

Reference Level for Potential Energy

The zero level for potential energy is an arbitrary reference point from which all other heights are measured. Choosing a convenient reference level, such as the lowest point or another significant point in the system, allows for consistent and meaningful comparison of gravitational potential energy values at different positions.

Conservation of Mechanical Energy and Conservative Forces

In systems involving only conservative forces like gravity, the total mechanical energy (the sum of kinetic and potential energies) remains constant. This principle of energy conservation allows one to relate the work done by gravity to changes in kinetic energy and facilitates solving problems where energy transforms from one form to another without losses due to friction or other non-conservative forces.

Transcript

00:01 Hello, everybody.

00:03 So we'll be solving another problem from chapter 8, and this will be question number six.

00:07 And it is once again a loop the loop.

00:10 Not once again, it is a loop the loop question.

00:13 So first of all, let's draw out what we have here, right? so here we go.

00:21 This is a loop -de -loop.

00:25 And so we have a block, and it is at point p.

00:30 This is point p.

00:33 And we have this loop -de -loop.

00:36 Well, actually, let me draw this loop -de -loop a little better.

00:41 So here we have a loop -de -loop.

00:45 Okay.

00:46 Now, this loop -de -loop has a radius of r, right? and the block, there's another block here at point q, and it is at a height of r.

01:02 Now, do we know what h, the height of the entire loop -de -loop is? well, it says that h is equal to five times the value of r, and it tells us that r is equal to 12 centimeters.

01:20 Therefore, h is equal to 60 centimeters.

01:26 Now, we need to do everything in meters.

01:30 So let's keep in mind that we need to convert centimeters to meters.

01:33 So 12 centimeters, right? now, we know what the conversion is.

01:40 So one meter is equal to 100 centimeters.

01:46 And then, so you have 0 .12 meters, and you have 60 centimeters, right? 1 meter is 100 centimeters, so it's equal to 0 .60 meters.

02:03 And now, let's move on to the question.

02:05 So the question wants to know how much work is the gravitational force doing on the block when it travels from the point p, which is the initial starting point, to different points along the loop -de -loop.

02:18 So a asks for the work done from p to q.

02:24 Very simple.

02:25 We know that work is equal to m .g .h.

02:30 H being the separation, right? so if we were to do this problem, well, we know what the mass of the block is because it gives us what it is.

02:39 So the mass of the block, let me write it out here, is actually 0 .032 kilograms.

02:46 So we know what the mass is.

02:48 We know what gravity is.

02:50 And what is h? well, h is the separation, right? so h is actually equal to h minus r.

02:58 Very, very simple.

03:00 So all you have to do is, you know, substitute the values.

03:03 So 0 .032 kilograms times 9 .8 meters per second squared times 0 .60 meters minus 0 .12 meters.

03:18 Now this is obviously why we convert centimeters to meters, right? because our gravitational constant is in meters per second.

03:25 If you were to put centimeters, you would either need to convert your gravitational constant or you know the 12 centimeters to meters...

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