A 20.0 kg solid cylinder, 0.25m in radius, and a 5.0 kg thin hollow sphere,
0.50 m in radius, are released from rest at the exact same time at the top of a
10m long ramp inclined at 30º.
a) What is the final velocity of the cylinder at the bottom of the ramp?
b) What is the final velocity of the sphere at the bottom of the ramp?
c) Which will reach the bottom first? By how many seconds?
To find the final velocity of the cylinder and the sphere at the bottom of the ramp, we can use the principle of conservation of energy. The initial potential energy at the top of the ramp will be converted into the final kinetic energy at the bottom. Let's solve these questions step by step:
a) Final velocity of the cylinder at the bottom of the ramp:
1. Calculate the initial potential energy of the cylinder at the top of the ramp:
Potential energy = mass * gravity * height
Potential energy = 20.0 kg * 9.8 m/s² * 10 m * sin(30º)
Note that we multiply by sin(30º) because the ramp is inclined at an angle of 30º.
2. Calculate the final kinetic energy of the cylinder at the bottom of the ramp:
Final kinetic energy = Initial potential energy
(1/2) * mass * final velocity² = mass * gravity * height * sin(30º)
3. Solve for the final velocity of the cylinder:
final velocity² = 2 * gravity * height * sin(30º)
final velocity = √(2 * 9.8 m/s² * 10 m * sin(30º))
b) Final velocity of the sphere at the bottom of the ramp:
Follow the same steps as above, but replace the mass of the cylinder with the mass of the sphere. Use the formula for the moment of inertia of a hollow sphere when calculating the initial potential energy:
Potential energy = (2/5) * mass * radius² * gravity * height * sin(30º)
c) To determine which object will reach the bottom first, we can compare their final velocities. The object with the higher final velocity will reach the bottom first.
To calculate the time it takes for each object to reach the bottom, we can use the equation of motion for constant acceleration:
s = ut + (1/2)at²
where s is the distance (10 m), u is the initial velocity (0 m/s), a is the acceleration (g * sin(30º)), and t is the time.
For each object, solve the equation for t, the time it takes to reach the bottom of the ramp.
Note: Make sure to convert the angle to radians in the equations, if necessary.