A bowling ball of mass 7.45 kg is rolling at 2.56 m/s along a level surface.

(a) Calculate the ball's translational kinetic energy.
J

(b) Calculate the ball's rotational kinetic energy.
J

(c) Calculate the ball's total kinetic energy.
J

(d) How much work would have to be done on the ball to bring it to rest?
J

A bowling ball of mass 7.50 kg is rolling at 2.84 m/s along a level surface.

(a) Calculate the ball's translational kinetic energy.
30.2
Correct: Your answer is correct.
J

(b) Calculate the ball's rotational kinetic energy.
12.1
Correct: Your answer is correct.
J

(c) Calculate the ball's total kinetic energy.

43

Correct: Your answer is correct.
J

(d) How much work would have to be done on the ball to bring it to rest?
-43
Correct: Your answer is correct.
J

(a) To calculate the ball's translational kinetic energy, we can use the formula:

Translational Kinetic Energy = (1/2) * mass * velocity^2

Given that the mass of the ball is 7.45 kg and the velocity is 2.56 m/s, we can substitute these values into the formula:

Translational Kinetic Energy = (1/2) * 7.45 kg * (2.56 m/s)^2

Simplifying the expression:

Translational Kinetic Energy = 1/2 * 7.45 kg * 6.5536 m^2/s^2

Translational Kinetic Energy ≈ 25.15112 J

Therefore, the ball's translational kinetic energy is approximately 25.15 J.

(b) To calculate the ball's rotational kinetic energy, we need to know the moment of inertia of the ball. Let's assume the moment of inertia of the ball is I.

Rotational Kinetic Energy = (1/2) * I * angular velocity^2

Since the ball is rolling, we can relate the angular velocity (ω) to the linear velocity (v) using the equation ω = v/r, where r is the radius of the ball. Let's assume the radius is given as R.

Angular velocity = v / R

Substituting the given values:

Angular velocity = 2.56 m/s / R

Rotational Kinetic Energy = (1/2) * I * (2.56 m/s / R)^2

We don't have enough information to calculate the moment of inertia (I) or the radius (R) of the ball, so we cannot determine the exact value of the rotational kinetic energy.

(c) The ball's total kinetic energy is the sum of its translational and rotational kinetic energy:

Total Kinetic Energy = Translational Kinetic Energy + Rotational Kinetic Energy

Since we don't know the value of the rotational kinetic energy, we cannot determine the total kinetic energy.

(d) The work done to bring the ball to rest will be equal to the initial kinetic energy of the ball, which is the sum of its translational and rotational kinetic energy.

Work = Translational Kinetic Energy + Rotational Kinetic Energy

Since we don't know the value of the rotational kinetic energy, we cannot determine the work required to bring the ball to rest.

To calculate the ball's translational kinetic energy, you need to use the formula:

Translational Kinetic Energy (KE_trans) = (1/2) * mass * velocity^2

(a) Plugging in the given values:

Mass (m) = 7.45 kg
Velocity (v) = 2.56 m/s

KE_trans = (1/2) * 7.45 kg * (2.56 m/s)^2
KE_trans = 24.077 J

Therefore, the ball's translational kinetic energy is 24.077 J.

To calculate the ball's rotational kinetic energy, you need to use the formula:

Rotational Kinetic Energy (KE_rot) = (1/2) * moment of inertia * angular velocity^2

However, we are not given the moment of inertia or angular velocity in this problem. So, we cannot calculate the rotational kinetic energy without that information.

(c) To find the ball's total kinetic energy, you simply add together the translational kinetic energy and the rotational kinetic energy:

Total Kinetic Energy (KE_total) = KE_trans + KE_rot

Since we cannot calculate the rotational kinetic energy, we cannot calculate the total kinetic energy.

(d) To calculate the work required to bring the ball to rest, we need to use the work-energy theorem:

Work (W) = Change in Kinetic Energy (ΔKE)

Since the ball is brought to rest, its final kinetic energy (KE_final) is zero. Therefore:

W = KE_final - KE_initial

Since KE_initial is the ball's initial kinetic energy (which we calculated in part a), we can substitute that value:

W = 0 J - 24.077 J
W = -24.077 J

Therefore, the work required to bring the ball to rest is -24.077 J. Note that the negative sign indicates that work is being done on the ball.

A- Translational KE, or Kt, = (1/2)mv^2. Plug in the mass and velocity.

B- Rotational KE, or Kr, = (1/2)Iw^2. Taking the ball's point of contact with the ground as the reference point, the moment of inertia I is (2/5)mr^2+mr^2 (due to the parallel axis theorem and that a solid sphere has a moment of inertia of (5/2)mr^2). Also, v=rw, so w=v/r, so w^2 = v^2/r^2. Plug this into the Kr equation and simplify and you will get:
Kr=(7/10)mv^2. plug in mass and velocity to get Kr.

C- The total kinetic energy is the answer to A plus the answer to B.

D- Due to the work-KE theorem we have W=Kfinal - Kinitial. If the Kfinal is zero because we are trying to stop the ball and when it is stopped KE=0, then W=0-Ki, W=-Ki. Plug in the total kinetic energy (answer to C) and you find the work needed to be done.