A hollow 0.358 kg sphere rolls without slipping down an inclined plane that makes an angle of 41.0o with the horizontal direction. The sphere is released from rest a distance 0.734 m from the lower end of the plane.
a. How fast is the hollow sphere moving as it reaches the end of the plane?
b. At the bottom of the incline, what fraction of the total kinetic energy of the hollow sphere is rotational kinetic energy?
This is what i got but i'm not sure its correct. Any help would be much appreciated!
a)
cons. of energy
1/2mv^2+1/2Iw^2=mgLsinθ
where
I=2/3mr^2
w=v/r
then
1/2mv^2+1/3mv^2=mgLsinθ
5/6mv^2=mgLsinθ
v=(6gLsinθ/5)^1/2
v=(6*9.8*0.734*sin41/5)^1/2 = 2.38 m/s
b)
K=1/2mv^2+1/3mv^2
K=5/6mv^2
Krotational=1/3mv^2
Krot/K=2/5=0.4
looks ok to me, I didn't do numbers.
To solve this problem, you correctly applied the law of conservation of energy to find the speed of the hollow sphere as it reaches the end of the plane.
For part A:
- Identify the initial and final positions of the sphere: The initial position is where the sphere is released from rest (0.734 m from the lower end of the plane), and the final position is at the end of the plane.
- Apply the conservation of energy equation: The sum of the initial kinetic energy (translational + rotational) and the initial potential energy (due to gravity) should be equal to the final potential energy (when the sphere reaches the end of the plane).
- Substitute the moments of inertia for a hollow sphere and the relationship between angular velocity and linear velocity for a rolling object.
- Rearrange the equation and solve for the linear velocity v.
For part B:
- Calculate the total kinetic energy of the hollow sphere at the bottom of the incline using the equation K = 1/2mv^2 + 1/3Iω^2, where I is the moment of inertia for a hollow sphere and ω is the angular velocity.
- Evaluate the ratio of rotational kinetic energy (1/3mv^2) to the total kinetic energy (K) to find the fraction.
Let's verify your calculations:
a) To find the speed of the hollow sphere as it reaches the end of the incline:
- Substitute the given values into the formula you derived: v = (6gLsinθ/5)^1/2.
- Calculate the value using g = 9.8 m/s^2, L = 0.734 m, and θ = 41.0°.
- The result is v = 2.38 m/s, which matches your calculation.
b) To determine the fraction of the total kinetic energy that is rotational kinetic energy:
- Use the equation Krot/K = 1/3mv^2 / (1/2mv^2 + 1/3mv^2).
- Simplify and evaluate the expression, which results in 2/5 or 0.4, which again matches your calculation.
Therefore, your answers for both parts are correct. Well done!