State what is meant by the moment of inertia of an object about an axis - AQA - A-Level Physics - Question 1 - 2020 - Paper 5

Start with the forces acting on the falling mass:

$F_{net} = m imes a = 0.5M imes g - T$

where $T$ is the tension in the string.

For the pulley, use the rotational form of Newton's second law:

$au = I imes ext{angular acceleration}$

Here, Torque $au = R imes T$, and the moment of inertia $I = 0.5MR^2$. Therefore:

R imes T = 0.5MR^2 imes rac{a}{R}

Simplifying gives:

$T = 0.25M imes a$

Substitute $T$ back into the net force equation:

$0.5M imes g - 0.25M imes a = 0.5M imes a$

Rearrange to solve for $a$:

$0.5g = 0.75M imes a$

This leads to:

a = rac{0.5g}{0.75} = 0.5g

For the spoked pulley, the same mass $M$ and radius $R$ are used. However, the moment of inertia may differ depending on the design of the pulley.

The load's acceleration in the case of the spoked pulley will still be affected by the mass of the load and the inertia of the pulley. Due to less mass being concentrated at the axis of rotation in spoked solutions, it is typically found that:

$a_{spoked} > a_{solid}$

This means the acceleration of the load will be greater in the second experiment, as there is less energy lost to rotational inertia.