Two blocks as shown in Figure P 10.71 are connected by a string of negligible mass passing over

Two blocks as shown in Figure P 10.71 are connected by a...

Two blocks as shown in Figure $\mathrm{P} 10.71$ are connected by a string of negligible mass passing over a pulley of radius 0.250 $\mathrm{m}$ and moment of inertia $I$ . The block on the frictionless incline is moving up with a constant acceleration of $2.00 \mathrm{m} / \mathrm{s}^{2} .$ (a) Determine $T_{1}$ and $T_{2},$ the tensions in the two parts of the string. (b) Find the moment of inertia of the pulley.

Key Concepts

Relationship Between Linear and Angular Acceleration

When a string does not slip on a pulley, the linear acceleration of the connected blocks is directly related to the angular acceleration of the pulley via the relation a = r?, where r is the radius of the pulley. This coupling allows the dynamics of the blocks (linear motion) and the pulley (rotational motion) to be linked, facilitating the determination of unknown variables such as tensions in the string and the moment of inertia of the pulley.

Moment of Inertia

The moment of inertia quantifies an object's resistance to changes in its rotational motion about a specific axis, serving a role analogous to mass in translational dynamics. It is a key parameter in problems involving rotational motion, as it influences how much torque is needed to achieve a certain angular acceleration. In the context of the pulley, it determines the relationship between the tensions in the string and the angular acceleration produced by the net torque.

Newton's Second Law

Newton's Second Law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. This principle is used to analyze the translational motion of the blocks, by accounting for all forces acting on each block, such as gravitational force components, tension, and normal force, to set up equations that ultimately allow the determination of the accelerations and tensions in the system.

Rotational Dynamics

Rotational dynamics involves applying Newton’s Second Law to rotating bodies, where the net torque on an object is equal to the product of its moment of inertia and its angular acceleration (? = I?). This concept is central to analyzing the pulley’s behavior, as the differences in tension on either side of the pulley produce a net torque that is resisted by the pulley's moment of inertia.

Transcript

00:01 In this exercise, we have the system shown here in the figure on the left that's composed by two masses, m1 that has a mass of 15 kilograms, and m2 that has a mass of 20 kilograms, that are connected by a string that passes through a pulley that has a radius of 0 .25 meters and a moment of inertia i.

00:27 And the acceleration of the system is two meters per second downwards for mass 2 and upwards from s1.

00:38 And our goal in question a is to find the tensions t1 and t2 in the string.

00:48 So in order to do that, i'm going to write the force equation for both m1 and m2.

00:58 So notice that acting upon m1 we have a gravitational force that has two components.

01:07 One of them is in the direction of motion, and the second one is perpendicular to the direction of motion.

01:16 Notice that the pull is not moving in the direction perpendicularly to the motion.

01:23 So this means that i'm only going to care about the forces that are that.

01:28 That are exerted on the direction of motion.

01:31 So for m1, we have the force that is pulling the block upwards is t1, and the one that's pushing it down is mg times sine of theta.

01:51 Theta is this angle here, which is equal to 37 degrees.

01:55 And this is equal to m times a.

02:00 So t1 is equal to m, actually here we should have m1.

02:06 So this is m1 times a plus g times sine of theta.

02:13 So this is 15 kilograms times a, which is 2 meters per second squared, plus g, which is 9 .8 meters per second squared, times the sign of 37.

02:30 And this is equal to 118 .5 newton.

02:38 Then we have to find t2.

02:41 Notice that the forces that act on t2 are just the gravitational force downwards and the tension force to 2...

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