A pulley of radius .10 m and having a rotational inertia of .30 kg-m^2 has a cord wrapped around it and attached to a 6.00 kg mass as shown. The pulley is free to rotate about a fixed axis through its center. The mass is allowed to fall from rest. What is the speed of the mass after it has fallen 2.0 m? Show all your work
To find the speed of the mass after it has fallen 2.0 m, we will first calculate the change in gravitational potential energy of the mass, and then use the principle of conservation of mechanical energy.
1. Calculate the change in gravitational potential energy:
The change in gravitational potential energy can be calculated using the formula:
ΔPE = m * g * h
Where:
ΔPE = change in gravitational potential energy
m = mass of the object (6.00 kg in this case)
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height the object has fallen (2.0 m in this case)
ΔPE = 6.00 kg * 9.8 m/s^2 * 2.0 m
ΔPE = 117.6 Joules
2. Calculate the change in kinetic energy:
Since the pulley is free to rotate and the cord is wrapped around it, some of the gravitational potential energy is converted into rotational kinetic energy of the pulley. To find the change in kinetic energy, we need to find the moment of inertia of the pulley:
I = 0.30 kg-m^2 (given)
The rotational kinetic energy can be calculated using the formula:
KE = (1/2) * I * ω^2
Where:
KE = rotational kinetic energy
I = moment of inertia
ω = angular velocity
2a. Calculate the angular velocity:
The angular velocity can be calculated using the formula:
ω = v / r
Where:
v = linear velocity of the falling mass
r = radius of the pulley
2.0 m = ω * 0.1 m
ω = 20 rad/s
2b. Calculate the change in kinetic energy:
KE = (1/2) * 0.30 kg-m^2 * (20 rad/s)^2
KE = 120 Joules
3. Calculate the final speed of the mass:
To find the final speed of the mass, we can equate the change in gravitational potential energy to the change in kinetic energy and solve for v (the final velocity):
ΔPE = KE
117.6 Joules = (1/2) * m * v^2
117.6 Joules = (1/2) * 6.00 kg * v^2
v^2 = 39.2 m^2/s^2
v = √(39.2 m^2/s^2)
v ≈ 6.26 m/s
Therefore, the speed of the mass after it has fallen 2.0 m is approximately 6.26 m/s.