A 0.75-mm-diameter solid sphere can be rotated about an axis through its center by a torque of 10.8 m∗Nm∗N which accelerates it uniformly from rest through a total of 170 revolutions in 14.0 ss .Part AWhat is the mass of the sphere?Express your answer to two significant figures and include the appropriate units.

To solve this problem, we need to use the equations of rotational motion and the moment of inertia for a solid sphere.

Step 1: Calculate the angular acceleration First, we need to convert the total revolutions to radians. There are 2π radians in one revolution, so 170 revolutions is 170*2π = 1068 radians.

The equation for angular displacement is θ = ωit + 0.5αt^2, where ωi is the initial angular velocity, t is the time, and α is the angular acceleration. Since the sphere starts from rest, ωi = 0, and the equation simplifies to θ = 0.5α*t^2.

We can rearrange this to solve for α: α = 2θ/t^2 = 21068/14^2 = 10.9 rad/s^2.

Step 2: Calculate the moment of inertia The torque τ is related to the angular acceleration by the equation τ = Iα, where I is the moment of inertia. We can rearrange this to solve for I: I = τ/α = 10.8/10.9 = 0.99 kgm^2.

Step 3: Calculate the mass The moment of inertia for a solid sphere rotating about an axis through its center is I = 2/5mr^2, where m is the mass and r is the radius. We can rearrange this to solve for m: m = I5/2r^2 = 0.995/(2*(0.75/2*10^-3)^2) = 111.3 kg.

So, the mass of the sphere is approximately 111 kg.