(hrw8c10p78) Two uniform solid spheres have the same mass, 1.65 kg, but one has a radius of 0.206 m while the other has a radius of 0.804 m. For each of the spheres, find the torque required to bring the sphere from rest to an angular velocity of 367 rad/s in 14.5 s. Each sphere rotates about an axis through its center. Torque on sphere with the smaller radius.

Complete question is:

Two uniform solid spheres have the same mass of 1.65 kg, but one has a radius of 0.206 m and the other has a radius of 0.804 m. Each can rotate about an axis through its center. (a) What is the magnitude τ of the torque required to bring the smaller sphere from rest to an angular speed of 367 rad/s in 14.5 s? (b) What is the magnitude F of the force that must be applied tangentially at the sphere’s equator to give that torque? What are the corresponding values of (c) τ and (d) F for the larger sphere?

A) τ = 0.709 N.m

B) F = 3.44 N

C) τ = 10.8 N.m

D) F = 13.43N

Explanation:

We are given;

Mass if each sphere = 1.65kg

Radius of the first sphere; r1 = 0.206m

Radius of second sphere; r2 = 0.804m

A) initial angular speed of smaller sphere; ω_i = 0 rad/s

Final angular speed of smaller sphere; ω_f = 367 rad/s

Time;t = 14.5 s

The constant angular acceleration is calculated from;

ω_f = ω_i + αt

367 = 0 + α(14.5)

Thus,

α = 367/14.5 = 25.31 rad/s²

The torque is given by the formula;

τ = Iα

Where τ is torque ; I is moment of inertia given as (2/5)Mr²

α is angular acceleration

Thus;

τ = (2/5)(1.65)(0.206)² x 25.31

τ = 0.709 N.m

B) The magnitude of the force that must be applied to give the torque τ is gotten from the formula;

τ = F•r•sin90°

0.709 = F x 0.206 x 1

F = 0.709/0.206

F = 3.44 N

C) Now for the larger sphere, we'll repeat the same procedure in a above. Thus;

The torque is given by the formula;

τ = Iα

Where τ is torque ; I is moment of inertia given as (2/5)Mr²

α is angular acceleration

Thus;

τ = (2/5)(1.65)(0.804)² x 25.31

τ = 10.8 N.m

D) Now for the larger sphere, we'll repeat the same procedure in b above. Thus, τ is gotten from the formula;

τ = F•r•sin90°

10.8 = F x 0.804 x 1

F = 10.8/0.804

F = 13.43N