A solid ball of mass 3 kg, rolls down a hill that is 7 meters high. What is the rotational KE at the bottom of the hill?
To calculate the rotational kinetic energy (KE) of the solid ball at the bottom of the hill, we need to know the moment of inertia and the angular velocity.
The moment of inertia (I) depends on the shape and mass distribution of the object. Assuming the solid ball is a uniform sphere, the moment of inertia can be calculated using the formula:
I = (2/5) * m * r^2
Where:
- I is the moment of inertia,
- m is the mass of the ball (3 kg),
- r is the radius of the ball.
The radius of the ball is not provided, so we cannot directly calculate the moment of inertia. However, assuming the ball is a perfect sphere, we know that the moment of inertia is directly proportional to the radius squared.
Now, let's move on to the angular velocity (ω) at the bottom of the hill. Since the ball is rolling down the hill, it undergoes both linear and rotational motion. The linear velocity of the ball depends on the height of the hill, but the angular velocity remains constant. So, we can calculate the angular velocity at the bottom of the hill using the following formula:
m * g * h = (1/2) * I * ω^2
Where:
- m is the mass of the ball (3 kg),
- g is the acceleration due to gravity (9.8 m/s^2),
- h is the height of the hill (7 m),
- I is the moment of inertia,
- ω is the angular velocity.
Rearranging the equation, we can solve for the angular velocity (ω):
ω = sqrt((2 * m * g * h) / I)
Substituting the known values, we can calculate the angular velocity.
Finally, we can determine the rotational kinetic energy (KE) using the formula:
KE rotational = (1/2) * I * ω^2
Now, let's calculate the moment of inertia, angular velocity, and rotational kinetic energy of the solid ball at the bottom of the hill.
loss of potential energy = m g h = 3*9.81* 7 = 206 Joules
= translational Ke + rotational Ke
= (1/2) m v^2 + (1/2)I omega^2
Note
I = (2/5) m r^2
v = omega r
so
206=(1/2)m v^2 + (1/2)(2/5)m r^2 v^2
412 = 1.4 *3 v^2
v = 9.9 m/s
then calculate (1/2) (2/5)(3) v^2