A sphere of mass 0.5kg and diameter 1m rolls without slipping with a constant velocity of 5m/s. Calculate itstotal energy
The total energy of the rolling sphere can be calculated by considering its translational kinetic energy and rotational kinetic energy.
Translational kinetic energy:
The translational kinetic energy (KE_trans) is given by the formula:
KE_trans = (1/2) * m * v^2
where m is the mass of the sphere and v is the velocity.
Given:
m = 0.5 kg
v = 5 m/s
KE_trans = (1/2) * 0.5 kg * (5 m/s)^2
= 2.5 J
Rotational kinetic energy:
The rotational kinetic energy (KE_rot) is given by the formula:
KE_rot = (1/2) * I * ω^2
where I is the moment of inertia of the sphere and ω is the angular velocity.
For a solid sphere rolling without slipping,
I = (2/5) * m * r^2
where r is the radius of the sphere.
Given:
m = 0.5 kg
diameter = 1 m
radius (r) = diameter/2 = 1 m/2 = 0.5 m
I = (2/5) * 0.5 kg * (0.5 m)^2
= 0.05 kg·m^2
From the velocity and the diameter, we can calculate the angular velocity (ω) using the formula for rolling motion:
v = ω * r
ω = v/r = 5 m/s / 0.5 m
= 10 rad/s
KE_rot = (1/2) * 0.05 kg·m^2 * (10 rad/s)^2
= 2.5 J
Total energy:
The total energy of the rolling sphere is the sum of its translational kinetic energy and rotational kinetic energy.
Total energy = KE_trans + KE_rot
= 2.5 J + 2.5 J
= 5 J
Therefore, the total energy of the rolling sphere is 5 Joules.