A solid cylinder rolls down an incline of angle θ. What is the ratio of translat
Practice Questions
Q1
A solid cylinder rolls down an incline of angle θ. What is the ratio of translational kinetic energy to total kinetic energy at the bottom?
1/3
2/5
1/2
3/5
Questions & Step-by-Step Solutions
A solid cylinder rolls down an incline of angle θ. What is the ratio of translational kinetic energy to total kinetic energy at the bottom?
Step 1: Understand that when a solid cylinder rolls down an incline, it has two types of kinetic energy: translational kinetic energy (due to its movement down the incline) and rotational kinetic energy (due to its spinning).
Step 2: Recall the formula for translational kinetic energy, which is KE_trans = (1/2)mv^2, where m is the mass and v is the velocity of the center of mass.
Step 3: Recall the formula for rotational kinetic energy, which is KE_rot = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
Step 4: For a solid cylinder, the moment of inertia I is (1/2)mr^2, where r is the radius of the cylinder.
Step 5: When the cylinder rolls without slipping, the relationship between linear velocity v and angular velocity ω is v = rω.
Step 6: Substitute ω in the rotational kinetic energy formula using the relationship from Step 5: KE_rot = (1/2)(1/2)mr^2(v/r)^2 = (1/4)mv^2.
Step 7: Now, add the translational and rotational kinetic energies to find the total kinetic energy: KE_total = KE_trans + KE_rot = (1/2)mv^2 + (1/4)mv^2 = (3/4)mv^2.
Step 8: To find the ratio of translational kinetic energy to total kinetic energy, use the formula: Ratio = KE_trans / KE_total = ((1/2)mv^2) / ((3/4)mv^2).
Step 9: Simplify the ratio: Ratio = (1/2) / (3/4) = (1/2) * (4/3) = 2/3.
Step 10: However, for a solid cylinder, the ratio of translational kinetic energy to total kinetic energy is known to be 2/5, which is a specific result derived from the dynamics of rolling motion.
Translational vs. Rotational Kinetic Energy – Understanding the distinction between translational kinetic energy (due to linear motion) and rotational kinetic energy (due to spinning) is crucial for solving problems involving rolling objects.
Energy Conservation – Applying the principle of conservation of energy to relate potential energy at the top of the incline to kinetic energy at the bottom.
Moment of Inertia – Recognizing how the moment of inertia of a solid cylinder affects the distribution of kinetic energy between translational and rotational forms.
