A solid cylinder rolls down an incline of height h. What is the speed of the cen

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Question: A solid cylinder rolls down an incline of height h. What is the speed of the center of mass at the bottom of the incline?

Options:

Correct Answer: √(3gh/2)

Solution:

Using conservation of energy, potential energy at the top equals kinetic energy at the bottom. For a solid cylinder, v = √(3gh/2).

A solid cylinder rolls down an incline of height h. What is the speed of the center of mass at the bottom of the incline?

A solid cylinder rolls down an incline of height h. What is the speed of the center of mass at the bottom of the incline?

Step 1: Understand that the solid cylinder starts at a height 'h' and has potential energy due to its height.
Step 2: Recognize that as the cylinder rolls down, this potential energy converts into kinetic energy.
Step 3: Remember the formula for potential energy (PE) at the top: PE = mgh, where 'm' is mass and 'g' is acceleration due to gravity.
Step 4: Know that when the cylinder reaches the bottom, all potential energy has turned into kinetic energy (KE). The total kinetic energy for a rolling object is KE = (1/2)mv^2 + (1/2)Iω^2, where 'I' is the moment of inertia and 'ω' is the angular velocity.
Step 5: For a solid cylinder, the moment of inertia I = (1/2)mr^2. Also, relate linear speed 'v' and angular speed 'ω' using the formula ω = v/r.
Step 6: Substitute I and ω into the kinetic energy formula to express everything in terms of 'v'.
Step 7: Set the potential energy equal to the total kinetic energy and solve for 'v'.
Step 8: After simplification, you will find that the speed of the center of mass at the bottom of the incline is v = √(3gh/2).

Conservation of Energy – The principle that the total energy in a closed system remains constant, allowing potential energy to convert into kinetic energy.
Rolling Motion – Understanding the relationship between translational and rotational motion for rolling objects, particularly how they affect kinetic energy.
Moment of Inertia – The solid cylinder's moment of inertia plays a crucial role in determining the distribution of mass and its effect on rolling motion.