a mass of m1 50 kg is placed on an inclined plane with inclination angle 30 and contact friction

Step 1:** Calculate the acceleration of the masses, which is equal to the angular acceleration o

A mass of m1 50 kg is placed on an inclined plane with...

A mass of m1 = 50 kg is placed on an inclined plane with inclination angle θ = 30◦ and contact friction coefficient µ = 0.2. The mass m1 is connected through a massless string to a mass m2 = 20 kg which hangs in the air. The string goes over a pulley made from a uniform disk of mass M = 4 kg and radius R = 0.1 m. The disk rotates around a horizontal axis without friction and the string does not slip on the disk. The pulley’s angular acceleration is α = 50 rads/s2 . Ignore air resistance on each object.

Transcript

00:01 All right, so let's say we have an inclined plane that is inclined at an angle of 30 degrees, and there is a mass attached to it, which is connected by a string to another mass, and the string runs over a pulley, and we're told the pulley's mass is four kilograms, and it has a radius 0 .1 meters.

00:21 Our hanging mass, we'll call this m2, we're told is 20 kilograms, and the mass on the incline is 50 kilograms.

00:34 And we're told that initially the pulley has an angular acceleration of 50 radians per second squared.

00:42 And the coefficient of friction for our hanging mass is 0 .2.

00:49 And so presumably what we want to know is what is the acceleration of either mass, which should be the same in magnitude, and then which direction is it? so we can find the acceleration in magnitude because it's going to be our angular velocity of our pulley times the radius of the pulley.

01:08 And so this should just be 5 meters per second squared.

01:13 So that's simple enough.

01:15 And then if we look at a free body diagram for mass 1, what we're going to have is it's on the plane.

01:21 The weight is going down.

01:24 And then it has a component of the weight along the plane like this.

01:30 A component perpendicular to the plane like this, and it's going to have the tension in this cable going up, and depending on which way it's moving, it's also going to encounter friction.

01:39 So if you look at mass 2, it's going to have, we'll call this t1.

01:44 This is going to have a different tension, t2 going up and the weight going down.

01:50 All right.

01:51 And if we look at the comparison that, or at the pulley, we have t1 minus t2 times the radius of the pulley is going to equal the torque on the pulley.

02:09 So this should be the moment of inertia, which is one -half m r squared times alpha.

02:20 So to figure out what direction it's going to move, let's compare m1, or sorry, i don't need to write this an angle...

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