A solid cylinder of mass M and radius R rolls down an incline of height h. What
Practice Questions
Q1
A solid cylinder of mass M and radius R rolls down an incline of height h. What is its speed at the bottom of the incline?
√(2gh)
√(gh)
√(4gh)
√(3gh)
Questions & Step-by-Step Solutions
A solid cylinder of mass M and radius R rolls down an incline of height h. What is its speed at the bottom of the incline?
Step 1: Understand that the solid cylinder starts at a height h on the incline.
Step 2: Recognize that at the top, the cylinder has potential energy due to its height.
Step 3: Remember that potential energy (PE) is calculated as PE = M * g * h, where M is mass and g is the acceleration due to gravity.
Step 4: As the cylinder rolls down, this potential energy converts into kinetic energy (KE) at the bottom.
Step 5: The total kinetic energy at the bottom includes both translational and rotational energy.
Step 6: The translational kinetic energy is KE_trans = (1/2) * M * v^2, where v is the speed.
Step 7: The rotational kinetic energy is KE_rot = (1/2) * I * ω^2, where I is the moment of inertia and ω is the angular velocity.
Step 8: For a solid cylinder, the moment of inertia I = (1/2) * M * R^2 and the relationship between linear speed v and angular speed ω is ω = v / R.
Step 9: Substitute ω into the rotational kinetic energy formula to get KE_rot = (1/2) * (1/2) * M * R^2 * (v/R)^2 = (1/4) * M * v^2.
Step 10: Combine the translational and rotational kinetic energies: KE_total = KE_trans + KE_rot = (1/2) * M * v^2 + (1/4) * M * v^2 = (3/4) * M * v^2.
Step 11: Set the potential energy equal to the total kinetic energy: M * g * h = (3/4) * M * v^2.
Step 12: Cancel M from both sides (assuming M is not zero): g * h = (3/4) * v^2.
Step 13: Solve for v^2: v^2 = (4/3) * g * h.
Step 14: Take the square root to find v: v = √((4/3) * g * h).
Step 15: Note that this is the speed at the bottom of the incline.
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 difference between translational and rotational kinetic energy, especially for rolling objects.
Inclined Plane Dynamics – Analyzing the motion of objects on inclined planes, including the effects of gravity and friction.
