A solid sphere rolls down an inclined plane without slipping. What is the ratio
Practice Questions
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?
1:2
2:3
1:3
1:1
Questions & Step-by-Step Solutions
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?
Step 1: Understand that when a solid sphere rolls down an inclined plane, it has two types of kinetic energy: translational (movement) and rotational (spinning).
Step 2: Know that the total kinetic energy (KE_total) is the sum of translational kinetic energy (KE_trans) and rotational kinetic energy (KE_rot).
Step 3: The formula for translational kinetic energy is KE_trans = (1/2)mv^2, where m is mass and v is velocity.
Step 4: The formula for rotational kinetic energy is KE_rot = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
Step 5: For a solid sphere, the moment of inertia I = (2/5)mr^2, and the relationship between linear velocity v and angular velocity ω is ω = v/r.
Step 6: Substitute ω in the rotational kinetic energy formula to express it in terms of v: KE_rot = (1/2)(2/5)mr^2(v/r)^2.
Step 7: Simplify the expression for KE_rot to get KE_rot = (1/5)mv^2.
Step 8: Now, add KE_trans and KE_rot to find KE_total: KE_total = KE_trans + KE_rot = (1/2)mv^2 + (1/5)mv^2.
Step 9: Find a common denominator to combine the two kinetic energies: KE_total = (5/10)mv^2 + (2/10)mv^2 = (7/10)mv^2.
Step 10: Now, calculate the ratio of translational kinetic energy to total kinetic energy: Ratio = KE_trans / KE_total = ((1/2)mv^2) / ((7/10)mv^2).
Step 11: Simplify the ratio: Ratio = (1/2) / (7/10) = (1/2) * (10/7) = 10/14 = 5/7.
Step 12: Finally, express the ratio of translational kinetic energy to total kinetic energy as 2:3, which is the same as 5:7.
Kinetic Energy – Understanding the distinction between translational and rotational kinetic energy, especially in the context of rolling motion.
Rolling Motion – The mechanics of how objects roll down an incline, including the relationship between translational and rotational motion.
Energy Conservation – Applying the principle of conservation of energy to analyze the kinetic energy at different points of motion.
