A small block with mass 0.0400 kg slides in a vertical...

A small block with mass 0.0400 kg slides in a vertical circle of radius $R =$ 0.500 m on the inside of a circular track. During one of the revolutions of the block, when the block is at the bottom of its path, point $A$, the normal force exerted on the block by the track has magnitude 3.95 N. In this same revolution, when the block reaches the top of its path, point $B$, the normal force exerted on the block has magnitude 0.680 N. How much work is done on the block by friction during the motion of the block from point $A$ to point $B$?

Key Concepts

Non-Conservative Forces

Non-conservative forces, such as friction, dissipate mechanical energy from a system, usually converting it to heat rather than storing it as potential energy. Unlike conservative forces where energy is conserved within a system, the work done by non-conservative forces alters the total mechanical energy, making them a critical component in understanding energy losses.

Circular Motion Dynamics

Circular motion dynamics involves the study of objects moving along a curved path where forces such as centripetal force keep the object in its circular path. In vertical circles, the interplay between gravitational, normal, and centrifugal forces leads to variations in the net force acting on the object at different positions, influencing speed and energy distribution throughout the motion.

Work-Energy Theorem

The work-energy theorem states that the total work done on an object is equal to its change in kinetic energy. This principle is invaluable in analyzing systems where energy is transferred or transformed, particularly when non-conservative forces like friction are present, as it helps quantify how energy losses manifest as reductions in kinetic energy.

Transcript

00:01 Problem 770, we have small block of a certain mass.

00:04 It slides in a vertical circle of radius half a meter inside the circle of track.

00:10 And during one of the resolutions of the block, a block at the bottom of the path, it has a normal force directed upward.

00:15 That's a contact force between the block and the surface that pushes the block away from the surface.

00:21 So it's directed upward.

00:22 And we have another normal force at point b that acts on the block pushing it down.

00:30 So that's in contact force between the surface and the block that keeps the block away from the surface.

00:38 And we're asked how much work is done on the block by friction during a motion of the block from point a to point b.

00:45 Okay.

00:46 So this immediately tells me by how much work is done by friction that i'm going to set up a conservation energy formula.

00:53 So initially, i have the kinetic energy due to block.

00:57 And we're going to define the bottom here to be the gravitational potential equal to zero so that there's going to be no gravitational potential at the point a.

01:08 Then we're told throughout the motion, there's going to be work done by friction.

01:13 So we're going to have minus work due to the friction.

01:19 Actually, i'm going to write that as plus.

01:21 And then that's going to be equal to the kinetic energy of point b.

01:28 So i'll write that as point two.

01:31 And then we're going to add the gravitational potential energy.

01:36 So that's going to be plus m times g times one.

01:39 One here because we have radius of half a meter, so 2r, the diameter from point a to point b, it's going to be one.

01:47 Okay.

01:48 So i have k1 and k2, i need to find in order to find what our work due to the friction is.

01:55 So in order to do that, i'm going to set up our sum forces of acting on these blocks at the respective.

02:03 Points and to find their accelerations each to do with points and then that will get me to the connect energy that's points too but i'll show you how i do that okay so our sum of forces at point a is going to be equal to our mass times our acceleration at that point now our acceleration at a point is going to be the centrifugal acceleration directed inward keeping the block moving in a circle.

02:36 So i write down what the acceleration is and that's going to be velocity at a squared over r.

02:44 And i notice that we have mass times velocity acceleration, velocity at point a squared and that looks somewhat similar to our, what we need for kinetic energy...

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