A 1.3-kg 10-cm-diameter solid sphere is rotating about its diameter at 77 rev/min.

(a) What is its kinetic energy?

(b) If an additional 5.0 mJ of energy are supplied to the rotational energy, what is the new angular speed of the ball?
rev/min

a) if it is rolling , it has rotational ke 1/2 I w^2, and translational KE 1/2 m v^2

To relate these, consider w*r=v

KE= 1/2 I v^2/r^2 + 1/2 m v^2

but I for a solid sphere is... 2/5 mr^2, so put that in for I, and compute.

To find the kinetic energy of the rotating sphere, we can use the formula:

Kinetic Energy (KE) = (1/2) * I * ω^2,

where I is the moment of inertia and ω is the angular velocity.

(a) First, we need to find the moment of inertia. For a solid sphere rotating about its diameter, the moment of inertia can be given as:

I = (2/5) * m * r^2,

where m is the mass of the sphere and r is the radius. Given that the mass is 1.3 kg and the diameter is 10 cm (which means the radius, r, is 5 cm or 0.05 m), we can calculate the moment of inertia:

I = (2/5) * 1.3 kg * (0.05 m)^2.

To find the angular velocity, we need to convert the given value of 77 rev/min to radians per second (rad/s). Since there are 2π radians in one revolution and 60 seconds in one minute, we have:

ω = (77 rev/min) * (2π rad/rev) * (1 min/60 s).

Now, substituting the values of I and ω into the formula for kinetic energy, we can find the answer to (a).

(b) To find the new angular speed of the ball if an additional 5.0 mJ of energy is supplied to the rotational energy, we first need to convert 5.0 mJ to joules (J). Since 1 mJ = 0.001 J, we have:

Energy supplied = 5.0 mJ = 0.005 J.

The total energy of the rotating sphere after the supplied energy is the sum of the original kinetic energy and the supplied energy:

Total energy = KE + Energy supplied.

Using the formula for kinetic energy mentioned earlier, we can rearrange it as:

KE = Total energy - Energy supplied.

Now, we can find the new angular speed by rearranging the kinetic energy formula and solving for ω:

KE = (1/2) * I * ω^2,

ω = √[(2 * KE) / I].

Substituting the new values of total energy and I into the formula, we can find the new angular speed in rev/min.