A disk and a ring of the same mass and radius are rolling without slipping down
Practice Questions
Q1
A disk and a ring of the same mass and radius are rolling without slipping down an incline. Which one will have a greater translational speed at the bottom?
Disk
Ring
Both have the same speed
Depends on the incline
Questions & Step-by-Step Solutions
A disk and a ring of the same mass and radius are rolling without slipping down an incline. Which one will have a greater translational speed at the bottom?
Step 1: Understand that both the disk and the ring have the same mass and radius.
Step 2: Know that they are rolling down an incline without slipping.
Step 3: Remember that potential energy (PE) at the top of the incline converts into kinetic energy (KE) at the bottom.
Step 4: Recognize that kinetic energy has two parts: translational kinetic energy (due to movement) and rotational kinetic energy (due to spinning).
Step 5: The formula for translational kinetic energy is KE_trans = 1/2 * m * v^2, where m is mass and v is translational speed.
Step 6: The formula for rotational kinetic energy is KE_rot = 1/2 * I * ω^2, where I is moment of inertia and ω is angular velocity.
Step 7: Understand that the moment of inertia (I) for a disk is lower than that for a ring of the same mass and radius.
Step 8: Since the disk has a lower moment of inertia, it can convert more of its potential energy into translational kinetic energy.
Step 9: Therefore, at the bottom of the incline, the disk will have a greater translational speed than the ring.
Moment of Inertia – The distribution of mass in an object affects how it rolls; a lower moment of inertia means more energy can be converted to translational motion.
Energy Conservation – The principle that potential energy is converted into kinetic energy as the objects roll down the incline.
Rolling Without Slipping – The condition where the object rolls down the incline without sliding, ensuring that both translational and rotational kinetic energy are considered.
