A uniform disc of mass M, radius R is attached to a frictionless horizontal axis. Its rim is wound with a light string and a tension T is applied to it, the angular acceleration will be?

Options

A. 2T/M
B. 2T/MR
C. T/MR
D. T/M

Answer

To find the angular acceleration of a uniform disc of mass M and radius R when a tension T is applied to a string wound around its rim, we can follow these steps:

### Step 1: Understand the system
We have a uniform disc with mass M and radius R. The disc is mounted on a frictionless horizontal axis, and a string is wound around its rim. When tension T is applied to the string, it creates a torque that causes the disc to rotate.

### Step 2: Calculate the torque due to the tension
The torque (τ) caused by the tension T can be calculated using the formula:
\[ \tau = T \cdot R \]
where R is the radius of the disc.

### Step 3: Relate torque to angular acceleration
Torque can also be expressed in terms of the moment of inertia (I) and angular acceleration (α) using the formula:
\[ \tau = I \cdot \alpha \]
For a uniform disc, the moment of inertia (I) is given by:
\[ I = \frac{1}{2} M R^2 \]

### Step 4: Set the two expressions for torque equal to each other
From Step 2 and Step 3, we can set the two expressions for torque equal:
\[ T \cdot R = \frac{1}{2} M R^2 \cdot \alpha \]

### Step 5: Solve for angular acceleration (α)
We can rearrange the equation to solve for angular acceleration (α):
\[ \alpha = \frac{2T}{M R} \]

### Final Answer
Thus, the angular acceleration of the disc is given by:
\[ \alpha = \frac{2T}{M R} \]