A hollow sphere rolls down a slope of height h. What fraction of its potential energy is converted into translational kinetic energy at the bottom?
Practice Questions
1 question
Q1
A hollow sphere rolls down a slope of height h. What fraction of its potential energy is converted into translational kinetic energy at the bottom?
1/3
1/2
2/3
1
For a hollow sphere, I = (2/3)mr^2. Using energy conservation, the translational kinetic energy is 2/3 of the potential energy at the top.
Questions & Step-by-step Solutions
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Q
Q: A hollow sphere rolls down a slope of height h. What fraction of its potential energy is converted into translational kinetic energy at the bottom?
Solution: For a hollow sphere, I = (2/3)mr^2. Using energy conservation, the translational kinetic energy is 2/3 of the potential energy at the top.
Steps: 10
Step 1: Understand that the hollow sphere has potential energy (PE) at the top of the slope due to its height (h). The formula for potential energy is PE = mgh, where m is mass, g is acceleration due to gravity, and h is height.
Step 2: When the hollow sphere rolls down the slope, it converts some of its potential energy into kinetic energy. This kinetic energy has two parts: translational kinetic energy (due to its movement) and rotational kinetic energy (due to its spinning).
Step 3: The moment of inertia (I) for a hollow sphere is given by the formula I = (2/3)mr^2, where r is the radius of the sphere.
Step 4: According to the conservation of energy, the total potential energy at the top (mgh) is equal to the total kinetic energy at the bottom (which is the sum of translational and rotational kinetic energy).
Step 5: The translational kinetic energy (KE_trans) can be expressed as KE_trans = (1/2)mv^2, where v is the velocity of the sphere at the bottom.
Step 6: The rotational kinetic energy (KE_rot) can be expressed as KE_rot = (1/2)Iω^2, where ω is the angular velocity. For rolling without slipping, v = rω.
Step 7: Substitute I into the rotational kinetic energy formula and relate ω to v to find the expression for KE_rot in terms of v.
Step 8: Combine the expressions for KE_trans and KE_rot to find the total kinetic energy at the bottom.
Step 9: Set the total kinetic energy equal to the potential energy at the top (mgh) and solve for the fraction of potential energy that is converted into translational kinetic energy.
Step 10: Conclude that for a hollow sphere, 2/3 of the potential energy is converted into translational kinetic energy at the bottom.
