6. A mass m1 = 4.50 kg is connected by a light string that...

6. A mass m1 = 4.50 kg is connected by a light string that passes over a pulley of mass m3 = 8.5 kg to a mass m2 = 8.00 kg sliding on a frictionless inclined surface that makes an angle of 29° with the horizontal (see figure). The coefficient of kinetic friction between the mass m1 and the horizontal surface is 0.16. There is no slippage between the string and the pulley. What is the magnitude of the acceleration of the system? (The moment of inertia of the pulley is ½Mr².)

Transcript

00:01 Hello, hello! in this question we're given this setup.

00:02 We have a block on this angled ramp and then that is connected over a non -massless pulley to another block which has a coefficient of friction.

00:12 We're given the masses and the angle and i've drawn the image here.

00:16 And so we're asked with this what is the acceleration of the system going to be? so in order to do this we're going to need to draw a free body diagram here on all of our parts.

00:25 So i'm going to have up here a mass 1.

00:27 I obviously have the weight of 1 pushing down.

00:30 I'm going to have the normal force pushing up and this is the normal force of 1.

00:35 And then because this is accelerating, it's connected to a string, it's going to have some tension.

00:40 I'll call it t1 going to the left.

00:43 And then i know that there's a force of friction and it's going to want to accelerate to the left.

00:47 So the force of friction is going to oppose that motion so i have the force of friction pushing to the right.

00:52 I then go over the pulley which is going to have two forces on it.

00:55 I'm going to have t1.

00:57 I want t1.

00:59 And then also this other force here which is going to be t2.

01:03 Those are both acting at a distance and then t2 is coming from this block down here which also has a weight going down which i'm going to break into the components here because that's going to be very handy.

01:15 That's going to be the angle theta there.

01:17 And then i also have my normal force pushing up so this will be normal force on 2.

01:21 And then there's no friction and there's nothing else pulling it down.

01:24 So the only thing accelerating this system is the force of gravity in the x direction of this block, block 2.

01:31 So there's two ways we can approach this.

01:32 We can write a sum of the forces for each of these and a sum of the torques for that one there.

01:37 And then we can do algebra to solve.

01:40 I'm going to opt to do that the sum of the forces of the system is going to equal the mass of the system times the acceleration of the system.

01:48 So my acceleration of the system is the sum of the forces.

01:51 What does that mean? well i know that this entire thing is going to accelerate in that direction because this is what's accelerating the system, the weight of that.

01:58 With that in mind any force that is pointing in the positive acceleration direction is going to be a positive force.

02:04 So for example i have the x component of my weight 2 which is going to be fg2 times the sine of theta in this case.

02:13 That's accelerating the system forward and then i have going in the other direction on that first on block number 2 i have t2 pulling back...

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