Question

(II) A 0.72-m-diameter solid sphere can be rotated about an axis through its center by a torque of 10.8 m • N which accelerates it uniformly from rest through a total of 160 revolutions in 15.0 s. What is the mass of the sphere

Answer 1

To find the mass of the sphere, we can use the formula for torque, which is given by T = Iα, where T is the torque, I is the moment of inertia, and α is the angular acceleration. We can calculate the moment of inertia of the sphere using the formula I = (2/5)MR^2, where M is the mass of the sphere and R is the radius. Finally, we can solve for the mass of the sphere using the formula T = Iα and the known values of torque and angular acceleration.

Step-by-step explanation:

To find the mass of the sphere, we can use the formula for torque, which is given by T = Iα, where T is the torque, I is the moment of inertia, and α is the angular acceleration. We can calculate the moment of inertia of the sphere using the formula I = (2/5)MR^2, where M is the mass of the sphere and R is the radius. Since the sphere is rotating uniformly, we can use the relation ω = αt, where ω is the angular velocity. We can find the angular velocity using the relation ω = 2πN, where N is the number of revolutions per unit time. Finally, we can solve for the mass of the sphere using the formula T = Iα and the known values of torque and angular acceleration.

Given:

Diameter of the sphere = 0.72 m

Torque = 10.8 m • N

Number of revolutions = 160

Time taken = 15.0 s

First, we need to convert the diameter to radius by dividing it by 2. So, the radius of the sphere is 0.72 m / 2 = 0.36 m.

Next, we can calculate the moment of inertia of the sphere using the formula I = (2/5)MR^2, where M is the mass of the sphere and R is the radius. Since the moment of inertia of a solid sphere is given by (2/5)MR^2, we can write it as (2/5) * M * (0.36 m)^2 = (2/5) * M * 0.1296 m^2.

Now, we can calculate the angular velocity using the formula ω = 2πN, where N is the number of revolutions per unit time. Since the number of revolutions is 160 and the time taken is 15.0 s, the angular velocity is ω = 2π * 160 / 15.0 rad/s.

Finally, we can solve for the mass of the sphere using the formula T = Iα and the known values of torque and angular acceleration. Since T = 10.8 m • N, α = ω / t, and t = 15.0 s, we have 10.8 m • N = (2/5) * M * 0.1296 m^2 * (2π * 160 / 15.0) / 15.0. Solving for M, we find that the mass of the sphere is approximately 0.69 kg.

Answer 2

`m = 23.3 kg`

Step-by-step explanation:

As we know that it will have constant torque on it

so the acceleration of the ball will be constant so here we can say that we can use kinematics equation

`\theta = \omega_i t + (1)/(2)\alpha t^2`

`160(2\pi) = 0 + (1)/(2)\alpha (15^2)`

`320 \pi = 112.5 \alpha`

so we have

`\alpha = (320\pi)/(112.5)`

`\alpha = 8.94 rad/s^2`

now we know that

`\tau = I \alpha`

`10.8 = I(8.94)`

`I = 1.21 kg m^2`

so we know that

`I = (2)/(5)mR^2`

here we know that

diameter = 0.72 m

so radius (R) = 0.36 m

`(2)/(5)m(0.36^2) = 1.21`

`m = 23.3 kg`