A 1.94 kg solid sphere (radius = 0.105 m) is released from rest at the top of a ramp and allowed to roll without slipping. The ramp is 0.710 m high and 5.17 m long. When the sphere reaches the bottom of the ramp, what is its total kinetic energy?
To find the total kinetic energy of the solid sphere at the bottom of the ramp, you need to consider both its translational and rotational kinetic energy.
1. Translational Kinetic Energy:
The translational kinetic energy is given by the formula: KE_trans = (1/2) * m * v^2, where m is the mass of the sphere and v is its linear velocity.
First, let's find the linear velocity of the sphere at the bottom of the ramp using conservation of energy.
The potential energy at the top of the ramp is equal to the sum of the translational and gravitational potential energy at the bottom:
m * g * h = (1/2) * m * v^2 + m * g * 0,
where g is the acceleration due to gravity and h is the height of the ramp.
Simplifying the equation gives:
v^2 = 2 * g * h
Substituting the given values:
v^2 = 2 * 9.8 m/s^2 * 0.710 m
Solving for v:
v = √(2 * 9.8 m/s^2 * 0.710 m)
Now that we have the linear velocity, we can compute the translational kinetic energy:
KE_trans = (1/2) * m * v^2
Substituting the given mass:
KE_trans = (1/2) * 1.94 kg * (√(2 * 9.8 m/s^2 * 0.710 m))^2
2. Rotational Kinetic Energy:
The rotational kinetic energy is given by the formula: KE_rot = (1/2) * I * ω^2, where I is the moment of inertia and ω is the angular velocity.
For a solid sphere rolling without slipping, the moment of inertia about its center of mass is given by: I = (2/5) * m * r^2, where r is the radius of the sphere.
Substituting the given values:
I = (2/5) * 1.94 kg * (0.105 m)^2
The angular velocity ω can be calculated using the linear velocity:
ω = v / r
Substituting the given radius and linear velocity:
ω = (√(2 * 9.8 m/s^2 * 0.710 m)) / 0.105 m
Now we can compute the rotational kinetic energy:
KE_rot = (1/2) * (2/5) * 1.94 kg * (0.105 m)^2 * (√(2 * 9.8 m/s^2 * 0.710 m) / 0.105 m)^2
Total Kinetic Energy:
The total kinetic energy is the sum of the translational and rotational kinetic energies:
KE_total = KE_trans + KE_rot
Substituting the calculated values:
KE_total = KE_trans + KE_rot
To find the total kinetic energy of the solid sphere at the bottom of the ramp, we need to calculate both its translational kinetic energy and rotational kinetic energy.
Let's begin with the translational kinetic energy. We can use the formula:
Translational Kinetic Energy = (1/2) * mass * velocity^2
Since the sphere is rolling without slipping, the velocity of the center of mass is related to its rotational motion. The relationship is given by:
velocity = radius * angular velocity
To find the angular velocity, we first need to determine the linear velocity at the bottom of the ramp. We can do this by using the conservation of energy.
The potential energy at the top of the ramp will be converted into both translational and rotational kinetic energy at the bottom. Therefore, we can write:
Potential Energy at top = Translational Kinetic Energy at bottom + Rotational Kinetic Energy at bottom
The potential energy at the top is given by:
Potential Energy at top = mass * gravity * height
Plugging in the values, where mass = 1.94 kg, gravity = 9.8 m/s², and height = 0.710 m, we can find the potential energy at the top.
Next, we need to find the rotational kinetic energy at the bottom. The formula for rotational kinetic energy is:
Rotational Kinetic Energy = (1/2) * moment of inertia * angular velocity^2
For a solid sphere rotating about its diameter, the moment of inertia is given by:
moment of inertia = (2/5) * mass * radius^2
Plugging in the values, we can calculate the moment of inertia.
Now, we can rearrange the potential energy equation to solve for the translational kinetic energy at the bottom. Then plug in the values of the potential energy, mass, and height, and solve for the translational kinetic energy.
Finally, to find the total kinetic energy, we add the translational kinetic energy and rotational kinetic energy of the sphere at the bottom of the ramp.