A solid sphere rolls down an inclined plane without slipping. What is the ratio of its translational kinetic energy to its total kinetic energy at the bottom of the incline?
Practice Questions
1 question
Q1
A solid sphere rolls down an inclined plane without slipping. What is the ratio of its translational kinetic energy to its total kinetic energy at the bottom of the incline?
1:2
2:3
1:3
1:1
The total kinetic energy is the sum of translational and rotational kinetic energy. For a solid sphere, the ratio of translational to total kinetic energy is 2:5, which simplifies to 2:3.
Questions & Step-by-step Solutions
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Q
Q: A solid sphere rolls down an inclined plane without slipping. What is the ratio of its translational kinetic energy to its total kinetic energy at the bottom of the incline?
Solution: The total kinetic energy is the sum of translational and rotational kinetic energy. For a solid sphere, the ratio of translational to total kinetic energy is 2:5, which simplifies to 2:3.
Steps: 10
Step 1: Understand that when a solid sphere rolls down an incline, it has two types of kinetic energy: translational (movement of the center of mass) and rotational (spinning around its axis).
Step 2: Know the formulas for kinetic energy: Translational kinetic energy (KE_trans) = (1/2)mv^2, where m is mass and v is velocity. Rotational kinetic energy (KE_rot) = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
Step 3: For a solid sphere, the moment of inertia (I) is (2/5)mr^2, where r is the radius of the sphere.
Step 4: When the sphere rolls without slipping, the relationship between linear velocity (v) and angular velocity (ω) is v = rω.
Step 5: Substitute ω in the rotational kinetic energy formula: KE_rot = (1/2)(2/5)mr^2(ω^2) = (1/5)mv^2 (since ω = v/r).
Step 6: Now, add the translational and rotational kinetic energies to find the total kinetic energy: Total KE = KE_trans + KE_rot = (1/2)mv^2 + (1/5)mv^2.
Step 7: Combine the terms: Total KE = (5/10)mv^2 + (2/10)mv^2 = (7/10)mv^2.
Step 8: Find the ratio of translational kinetic energy to total kinetic energy: Ratio = KE_trans / Total KE = ((1/2)mv^2) / ((7/10)mv^2).
Step 9: Simplify the ratio: The mv^2 cancels out, giving Ratio = (1/2) / (7/10) = (1/2) * (10/7) = 10/14 = 5/7.
Step 10: The final ratio of translational kinetic energy to total kinetic energy is 5:7.
