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A uniform disc with radius r and mass M stands upright on a rough surface, and a thin thread is wrapped around its circumference. The thread is pulled with a constant force T along a line tangential to the surface. Calculate the acceleration of the centre of the disc and the angular acceleration of the disc, if it rolls without slipping.

Sagot :

Answer:

Acceleration of the center of the disc:

a = (T - Mg + N) / M

Angular acceleration of the disc:

α = 2 × T / (M × r)

Since N = Mg (equilibrium) and g = 9.8 m/s^2 (standard gravity), we can simplify the expressions:

a = (T - Mg + Mg) / M = T / M

α = 2 × T / (M × r)

So, the answers are:

a = T / M

α = 2 × T / (M × r)

Please note that these are general expressions, and you'll need to plug in specific values for T, M, and r to get numerical answers.

Explanation:

classic problem in mechanics!

Let's break it down step by step:

1. ** Forces acting on the disc **:

   - Force of gravity (Mg) acting downwards

   - Normal force (N) acting upwards from the rough surface

   - Force of tension (T) in the thread, acting tangentially

   - Friction force (f) acting opposite to the motion

2. ** Motion of the disc **:

   - The disc rolls without slipping, so the point of contact with the surface is at rest.

   - The center of the disc moves with acceleration (a)

   - The disc rotates with angular acceleration (α)

3. ** Equations of motion **:

   - For linear motion: Mg - N + T = Ma (since f = 0, no slipping)

For rotational motion: T × r = I × α (where I is the moment of inertia of the disc)

4. ** Moment of inertia **:

   - For a uniform disc: I = (1/2) × M × r^2

5. ** Solving for a and α **:

   - From the linear motion equation: a = (T - Mg + N) / M

   - From the rotational motion equation: α = T × r / I = T × r / (0.5 × M × r^2) = 2 × T / (M × r)

So, the acceleration of the center of the disc is:

a = (T - Mg + N) / M

And the angular acceleration of the disc is:

α = 2 × T / (M × r)

Note that N = Mg if the disc is in equilibrium, and f = 0 since it rolls without slipping. You can plug in the values of T, M, r, and g to find the numerical values of a and α.

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