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 at the bottom of the incline? (2021)
Practice Questions
1 question
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 at the bottom of the incline? (2021)
√(2gh)
√(3gh/2)
√(gh)
√(4gh/3)
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, mgh = (1/2)mv^2 + (1/5)mv^2, leading to v = √(10gh/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 at the bottom of the incline? (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, mgh = (1/2)mv^2 + (1/5)mv^2, leading to v = √(10gh/7).
Steps: 10
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 energy is kinetic energy, which has two parts: translational kinetic energy (due to the motion of the center of mass) and rotational kinetic energy (due to the spinning of the sphere).
Step 3: Write the formula for translational kinetic energy, which is KE_trans = (1/2)mv^2, where v is the speed of the center of mass.
Step 4: Write the formula for rotational kinetic energy for a solid sphere, which is KE_rot = (1/5)mv^2.
Step 5: Combine the two kinetic energy formulas to get the total kinetic energy at the bottom: KE_total = KE_trans + KE_rot = (1/2)mv^2 + (1/5)mv^2.
Step 6: Set the initial potential energy equal to the total kinetic energy: mgh = (1/2)mv^2 + (1/5)mv^2.
Step 7: Factor out the common term 'm' from both sides of the equation, which simplifies to gh = (1/2)v^2 + (1/5)v^2.
Step 8: Combine the fractions on the right side: (1/2 + 1/5) = (5/10 + 2/10) = (7/10), so gh = (7/10)v^2.
Step 9: Solve for v^2 by multiplying both sides by (10/7): v^2 = (10/7)gh.
Step 10: Take the square root of both sides to find v: v = √(10gh/7).
