3. This problem about block and pulley system has two parts....

3. This problem about block and pulley system has two parts. Both pulleys are frictionless of the same mass M and radius R. A block of mass m1 is at rest on an inclined plane that makes an angle ? with the horizontal. The coefficient of static friction between the block and the incline surface is ?s. Part 1: The system is in an equilibrium. Determine the maximum value of m2 for which block 1 can remain at rest. Express your answer in terms of m1, ?, ?s and g (don't forget to draw your forces) Block 1 m1 ? Pulley 2 m2

Transcript

00:01 So let's say we have an incline plane, a certain angle of theta, and we have a block on here, and it's attached to a pulley, which is then attached to another pulley, which is then attached to the ceiling, and this movable pulley is hatched to a hanging mass.

00:19 And there's friction between the block and the incline like this.

00:24 And so we want to know what is the maximum mass into that keeps block one stationery? so if we look at the free body diagram of block one, we're going to have the tension going this way.

00:40 We're going to have the frictional force going this way, presumably.

00:46 The weight goes down.

00:48 The component of the weight along the plane is m .g.

00:50 Sign theta and this is m g cosine theta this is like our normal force up in this direction and then for block two what do we have we basically just have the tension going up but the tension here is really like t over two on each side so i should write this as t over two just because they're the tensions are equal everywhere and then we have the weight going down so what we have is like our new first law for this says m1a is going to equal like t over two or perhaps you want to call it t and just call these t as well.

01:32 Just note that you're going to have a relationship where the tension on each side is equal.

01:37 That's what each of those tensions is supporting the weight.

01:39 So we'll have m1a equals t minus m1g sine theta.

01:48 And the fictional force should be like this.

01:59 And actually, so what i forgot to mention is that the pulleys actually have masses as well.

02:04 So let's call this t1, let's call this t2, and let's call this t3.

02:11 So this is t1, this is two, this is three.

02:15 This is our pulley.

02:17 Both pulleys have the same mass and the same radius.

02:20 So what we'll have is like t2 minus t1 times the radius of the pulley.

02:26 That's the torque generated on the pulley.

02:28 Is just going to be the moment of inertia, which is like 1 .5m .r...

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