A 6.25 kg solid ball with a radius of 0.185 m rolls without slipping at 3.55 m/s. What is its total kinetic energy?
To find the total kinetic energy of the rolling ball, we need to consider both its translational kinetic energy and its rotational kinetic energy.
1. Translational kinetic energy:
The translational kinetic energy of an object is given by the formula: KE_trans = (1/2) * m * v^2
where m is the mass of the object and v is its linear velocity.
In this case, the mass of the ball is 6.25 kg and its linear velocity is 3.55 m/s. Plugging these values into the formula, we have:
KE_trans = (1/2) * 6.25 kg * (3.55 m/s)^2
2. Rotational kinetic energy:
The rotational kinetic energy of a rolling object is given by the formula: KE_rot = (1/2) * I * ω^2
where I is the moment of inertia of the object and ω is its angular velocity.
For a solid ball, the moment of inertia is given by: I = (2/5) * m * r^2
where r is the radius of the ball.
In this case, the radius of the ball is 0.185 m. Plugging this value into the moment of inertia formula, we have:
I = (2/5) * 6.25 kg * (0.185 m)^2
The angular velocity ω of a rolling object can be calculated using the formula: ω = v / r
where v is the linear velocity and r is the radius.
In this case, the linear velocity of the ball is 3.55 m/s. Plugging this value into the formula, we have:
ω = 3.55 m/s / 0.185 m
Now that we have the moment of inertia and the angular velocity, we can calculate the rotational kinetic energy using the formula mentioned earlier:
KE_rot = (1/2) * I * ω^2
Finally, to find the total kinetic energy, we need to add the translational and rotational kinetic energies together:
Total KE = KE_trans + KE_rot
Now you can follow these steps and plug in the values to find the total kinetic energy of the rolling ball.
It consists of two parts:
(a) The translational KE,
(1/2) M V^2,
which is due to the motion of the center of mass at velocity V = 3.55 m/s, and
(b) The rotational KE,
(1/2)*I*w^2
where w is the angular velocity, V/R, and I is the moment of inertia, which for a solid sphere is (2/5)M*R^2
Put then all together and you get
KE(total) = (1/2 + 1/5) M V^2
= (7/10)*M V^2