Question

Two identical blocks, each of mass $M$, are connected by a light string over a frictionless pulley of radius $R$ and rotational inertia $I$ (Fig. 9-55). The string does not slip on the pulley, and it is not known whether or not there is friction between the plane and the sliding block. When this system is released, it is found that the pulley turns through an angle $\theta$ in time $t$ and the acceleration of the blocks is constant. (a) What is the angular acceleration of the pulley? ( $b$ ) What is the acceleration of the two blocks? ( $c$ ) What are the tensions in the upper and lower sections of the string? All answers are to be expressed in terms of $M,I,R,\theta ,g$, and $t$.

Short answer

The angular acceleration is $2\theta /t^2$, the acceleration of the blocks is $2R\theta /t^2$, and the tensions in the upper and lower sections of the string are $Mg-2MR\theta /t^2$ and $Mg+2MR\theta /t^2$ respectively.

Step-by-step solution

Find the angular acceleration of the pulley

The angular acceleration $\alpha$ of the pulley is equal to the angle $\theta$ divided by half the time squared. This comes from the equation $\theta =\frac{1}{2}\alpha t^2$. Hence, $\alpha =2\theta /t^2$.

Find the acceleration of the two blocks

The acceleration $a$ of the blocks is related to the angular acceleration of the pulley through the radius of the pulley. This is because the string does not slip on the pulley. Hence, acceleration $a=\alpha R$. Substituting $\alpha$ from Step 1, $a=2R\theta /t^2$.

Find the tensions in the upper and lower sections of the string

The tension $T_1$ in the upper section of the string equals the force experienced by the falling block, which is its weight minus its force due to acceleration. Therefore, $T_1=Mg-Ma$. Substituting $a$ from Step 2, $T_1=Mg-2MR\theta /t^2$.The tension $T_2$ in the lower section of the string is the sum of the weight of the block on the inclined plane and the force due to acceleration in the direction of motion. Therefore, $T_2=Mg+Ma$. Substituting $a$ from Step 2, $T_2=Mg+2MR\theta /t^2$.

Key concepts

Angular Acceleration

Angular acceleration is a measure of how quickly an object's rotation is speeding up or slowing down. It represents the change in angular velocity over time. In this exercise, we find the angular acceleration $\alpha$ using the relationship between the angle the pulley turns $\theta$, and the time $t$.

• Formula used: $\theta =\frac{1}{2}\alpha t^2$
• Rearranging gives: $\alpha =\frac{2\theta }{t^2}$

This formula shows that to calculate angular acceleration, you divide the angle by half the square of the time it takes to rotate through that angle. Since the problem states the angle and time, we can directly find $\alpha$ by substitution.
Understanding angular acceleration is crucial, especially in systems where rotational motion is involved, as it helps predict how fast a pulley system will turn.

Frictionless Pulley

A frictionless pulley is an idealized concept where the pulley rotates without any resistance from friction. This assumption simplifies calculations by allowing us to disregard frictional losses that usually occur in real-life scenarios. In this exercise,

• The string does not slip over the pulley.
• The absence of friction ensures all force is transmitted effectively between the pulley and the string.

This leads to a direct relationship between the linear motion of the string and the rotational motion of the pulley.
Consequently, the linear acceleration of the string directly influences the angular acceleration of the pulley, which can be advantageous in solving the problem without factoring in additional losses.

Rotational Inertia

Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to changes in its rotation. This value depends on the object's mass and how that mass is distributed relative to the axis of rotation.

• Rotational inertia $I$ is a crucial quantity in determining how quickly an object like a pulley can accelerate under an applied torque.

When dealing with pulleys, rotational inertia plays a significant role because it dictates how much torque is required to achieve a certain angular acceleration.
Therefore, knowing $I$ helps us understand the dynamics of the system in this exercise, especially when combined with Newton's Laws of Motion.

Newton's Laws of Motion

Newton's Laws of Motion form the foundation for understanding the motion of objects. In this pulley system, these laws are utilized to better understand and calculate motion and forces:

• First Law (Inertia): The blocks and pulley remain at rest until a net external force acts on them.
• Second Law (F=ma): We derive the equations for tension, which include both gravitational forces and acceleration.
• Third Law (Action-Reaction): The string exerts equal and opposite forces on blocks connected across the pulley.

Using these principles, the tension in the string sections was derived. The tensions in the upper $T_1$ and lower $T_2$ sections depend on the gravitational force $Mg$ and the product of mass and block acceleration $Ma$. Newton’s laws help explain the interplay of forces and derived quantities such as acceleration and tension in this context.