A solid sphere of uniform density starts from rest and rolls without slipping a distance of d = 4.4 m down a q = 20° incline. The sphere has a mass M = 4.6 kg and a radius R = 0.28 m. What is the translational kinetic energy of the sphere when it reaches the bottom of the incline?
KEtran = ?
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To find the translational kinetic energy of the sphere when it reaches the bottom of the incline, we need to know the formula for translational kinetic energy.
The translational kinetic energy (KEtran) of an object is given by the formula:
KEtran = 0.5 * mass * velocity^2
However, to find the velocity of the sphere at the bottom of the incline, we need to use the principles of rotational motion since the sphere is rolling without slipping.
The velocity of the sphere can be determined using the equation relating linear velocity (v) and angular velocity (ω) for an object rolling without slipping:
v = ω * radius
The angular velocity (ω) can be determined by using the relationship between angular velocity, linear velocity, and the radius of the sphere:
ω = v / R
Now, to find the angular velocity (ω), we can use the rotational motion equations for an object rolling without slipping:
ω^2 = (2 * acceleration * distance) / (radius * (1 + (1 / inertia)))
Where:
- acceleration is the acceleration due to gravity on the incline (g * sin(q))
- distance is the distance traveled down the incline (d)
- radius is the radius of the sphere (R)
- inertia is the moment of inertia of the sphere (2/5 * M * R^2)
By substituting the given values into the equations, we can solve for the translational kinetic energy (KEtran).
Let's calculate it step by step:
1. Determine the angular velocity (ω):
First, calculate the acceleration due to gravity on the incline, which is given by:
acceleration = g * sin(q)
where g is the acceleration due to gravity (approximately 9.8 m/s^2) and q is the angle of the incline (20°).
Substituting the values:
acceleration = 9.8 m/s^2 * sin(20°)
Next, calculate the moment of inertia of the sphere:
inertia = 2/5 * M * R^2
Substituting the given values:
inertia = 2/5 * 4.6 kg * (0.28 m)^2
2. Calculate the angular velocity (ω):
Use the equation:
ω^2 = (2 * acceleration * distance) / (radius * (1 + (1 / inertia)))
Substituting the calculated values:
ω^2 = (2 * acceleration * distance) / (radius * (1 + (1 / inertia)))
3. Calculate the linear velocity (v):
Use the relationship:
v = ω * radius
Substituting the calculated values:
v = ω * radius
4. Calculate the translational kinetic energy (KEtran):
Use the formula:
KEtran = 0.5 * mass * v^2
Substituting the given mass and calculated velocity:
KEtran = 0.5 * mass * v^2
By following these steps and performing the necessary calculations, you can find the translational kinetic energy (KEtran) of the sphere when it reaches the bottom of the incline.