A spherical bowling ball radius 0.1 m and mass4.0 kg is rolling without slipping to ward the ball rack assume h=0.45m the liners sped at the bottom of the step is 2.85 m/s
A find moment of interia of the ball
B find the total kinetic energy st the bottom of step
C find the total kinetic energy at the top of the step
A) To find the moment of inertia of the ball, we can use the formula for the moment of inertia of a solid sphere, which is given by:
I = (2/5) * m * r^2
Where:
I = moment of inertia
m = mass of the sphere
r = radius of the sphere
Given:
m = 4.0 kg
r = 0.1 m
Substituting the given values into the formula, we can calculate the moment of inertia:
I = (2/5) * (4.0 kg) * (0.1 m)^2
I = (2/5) * (4.0 kg) * (0.01 m^2)
I = (2/5) * (0.04 kg m^2)
I = 0.08 kg m^2
The moment of inertia of the ball is 0.08 kg m^2.
B) To find the total kinetic energy at the bottom of the step, we can use the formula for rotational kinetic energy, which is given by:
KE_r = (1/2) * I * ω^2
Where:
KE_r = rotational kinetic energy
I = moment of inertia
ω = angular velocity
Given:
I = 0.08 kg m^2 (from part A)
ω = ω_linear / r (since the ball is rolling without slipping, linear velocity and angular velocity are related)
The linear velocity, v_linear, can be found using the formula:
v_linear = r * ω_linear
Given:
r = 0.1 m
v_linear = 2.85 m/s
Substituting the given values, we can calculate ω_linear:
2.85 m/s = 0.1 m * ω_linear
ω_linear = 2.85 m/s / 0.1 m
ω_linear = 28.5 rad/s
Now, we can find the angular velocity, ω, using:
ω = ω_linear / r
ω = 28.5 rad/s / 0.1 m
ω = 285 rad/s
Substituting the moment of inertia and angular velocity into the formula for rotational kinetic energy, we can calculate the total kinetic energy at the bottom of the step:
KE_r = (1/2) * (0.08 kg m^2) * (285 rad/s)^2
KE_r = (1/2) * (0.08 kg m^2) * (81225 rad^2/s^2)
KE_r = 3258.0 J
The total kinetic energy at the bottom of the step is 3258.0 Joules.
C) To find the total kinetic energy at the top of the step, we can assume that there is no loss of energy due to friction or other factors. Therefore, the total kinetic energy at the top of the step will be equal to the total kinetic energy at the bottom of the step, which is 3258.0 Joules.