A bowling ball (solid sphere of mass = 7.27 kg and radius = 12.7 cm) is rolling on level ground at a constant angular speed of 55 rad/s. What is the total kinetic energy of the ball?
To find the total kinetic energy of the ball, we need to consider both the translational kinetic energy and the rotational kinetic energy.
1. Translational kinetic energy (KE_trans):
The translational kinetic energy is given by the formula:
KE_trans = (1/2) * mass * velocity^2
In this case, since the ball is rolling on level ground with a constant angular speed, it has both linear velocity and angular velocity.
The linear velocity (v) can be calculated using the formula:
v = radius * angular velocity
Substituting the given values:
v = 0.127 m * 55 rad/s = 6.985 m/s
Now we can calculate the translational kinetic energy:
KE_trans = (1/2) * 7.27 kg * (6.985 m/s)^2
2. Rotational kinetic energy (KE_rot):
The rotational kinetic energy is given by the formula:
KE_rot = (1/2) * moment of inertia * angular velocity^2
The moment of inertia (I) for a solid sphere is given by the formula:
I = (2/5) * mass * radius^2
Substituting the given values:
I = (2/5) * 7.27 kg * (0.127 m)^2
Now we can calculate the rotational kinetic energy:
KE_rot = (1/2) * [(2/5) * 7.27 kg * (0.127 m)^2] * (55 rad/s)^2
Finally, we can find the total kinetic energy by adding the translational and rotational kinetic energies:
Total KE = KE_trans + KE_rot
To calculate the total kinetic energy of the bowling ball, we need to consider both its linear and rotational kinetic energy.
1. Linear kinetic energy (KE_linear):
The linear kinetic energy is given by the formula:
KE_linear = (1/2) * mass * velocity^2
The velocity of the bowling ball can be calculated using the angular speed and the radius:
velocity = angular speed * radius
Plugging in the values:
velocity = 55 rad/s * 0.127 m (converting radius to meters)
velocity = 7.035 m/s
Now, we can calculate the linear kinetic energy:
KE_linear = (1/2) * 7.27 kg * (7.035 m/s)^2
2. Rotational kinetic energy (KE_rotational):
The rotational kinetic energy is given by the formula:
KE_rotational = (1/2) * moment of inertia * angular speed^2
The moment of inertia for a solid sphere is defined as:
moment of inertia = (2/5) * mass * radius^2
Plugging in the values:
moment of inertia = (2/5) * 7.27 kg * (0.127 m)^2
Now, we can calculate the rotational kinetic energy:
KE_rotational = (1/2) * [(2/5) * 7.27 kg * (0.127 m)^2] * (55 rad/s)^2
3. Total kinetic energy:
The total kinetic energy is the sum of the linear and rotational kinetic energy:
Total KE = KE_linear + KE_rotational
Plugging in the values calculated above, we can find the total kinetic energy of the ball.