A solid sphere of mass M and radius R rolls without slipping down an inclined plane of height h. What is the speed of the center of mass of the sphere when it reaches the bottom? (2021)
Practice Questions
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Q1
A solid sphere of mass M and radius R rolls without slipping down an inclined plane of height h. What is the speed of the center of mass of the sphere when it reaches the bottom? (2021)
√(2gh)
√(5gh/7)
√(3gh/5)
√(gh)
Using conservation of energy, potential energy at the top = kinetic energy at the bottom. The total kinetic energy is the sum of translational and rotational kinetic energy. Thus, v = √(5gh/7).
Questions & Step-by-step Solutions
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Q
Q: A solid sphere of mass M and radius R rolls without slipping down an inclined plane of height h. What is the speed of the center of mass of the sphere when it reaches the bottom? (2021)
Solution: Using conservation of energy, potential energy at the top = kinetic energy at the bottom. The total kinetic energy is the sum of translational and rotational kinetic energy. Thus, v = √(5gh/7).
Steps: 9
Step 1: Identify the initial energy of the sphere at the top of the incline. This is the potential energy, which is given by the formula PE = Mgh, where M is the mass, g is the acceleration due to gravity, and h is the height.
Step 2: Identify the final energy of the sphere at the bottom of the incline. This is the kinetic energy, which consists of two parts: translational kinetic energy (KE_trans = 1/2 Mv^2) and rotational kinetic energy (KE_rot = 1/2 Iω^2).
Step 3: For a solid sphere, the moment of inertia I is given by I = 2/5 MR^2. Also, since the sphere rolls without slipping, the relationship between linear speed v and angular speed ω is ω = v/R.
Step 4: Substitute the expression for I and ω into the rotational kinetic energy formula: KE_rot = 1/2 (2/5 MR^2)(v/R)^2 = 1/5 Mv^2.
Step 5: Combine the translational and rotational kinetic energies: Total KE = KE_trans + KE_rot = 1/2 Mv^2 + 1/5 Mv^2 = (5/10 Mv^2 + 2/10 Mv^2) = 7/10 Mv^2.
Step 6: Set the initial potential energy equal to the total kinetic energy at the bottom: Mgh = 7/10 Mv^2.
Step 7: Cancel M from both sides of the equation: gh = 7/10 v^2.
Step 8: Solve for v^2: v^2 = (10/7)gh.
Step 9: Take the square root to find v: v = √(10gh/7).
