Multiple Choice
When you release a simple 100-g yo-yo from rest, it falls and rolls, unwinding the light string around its cylindrical shaft, which is 2 cm in radius. If the yo-yo can be modeled after a solid disc, calculate its linear acceleration.
13. Rotational Inertia & Energy
Rotational Dynamics with Two Motions
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Two blocks of masses m1 and m2 are both attached to a long, light rope that is wrapped several times around a pulley, as shown below. The pulley has mass M and radius R, can be modeled as a solid cylinder, and is free to rotate about a fixed, frictionless axis perpendicular to itself and through its center. When the block is released from rest, it begins to fall, causing the pulley to unwind without slipping. Derive an expression for the angular acceleration of the pulley.
A
α = g(m1−m2) / (m1+m2)
B
α = (g/R)(m1−m2) / (m1+m2+M)
C
α = (g/R)(m1+m2) / (m1+m2+M)
D
α = (g/R)(m1−m2) / (m1+m2+M/2)
Verified step by step guidance1
Start by analyzing the forces acting on the system. The two blocks, m1 and m2, are subject to gravitational forces, which are m1 * g and m2 * g respectively, where g is the acceleration due to gravity.
Consider the torque exerted on the pulley due to the tension in the rope. The torque τ is given by τ = R * (T1 - T2), where T1 and T2 are the tensions in the rope on either side of the pulley, and R is the radius of the pulley.
Apply Newton's second law for rotation, which states that the net torque τ is equal to the moment of inertia I of the pulley times its angular acceleration α: τ = I * α. For a solid cylinder, the moment of inertia I is (1/2) * M * R^2, where M is the mass of the pulley.
Relate the linear acceleration a of the blocks to the angular acceleration α of the pulley using the relation a = R * α. This is because the rope unwinds without slipping, meaning the linear speed of the blocks is equal to the tangential speed of the pulley.
Combine these equations to solve for α. Substitute the expression for torque and moment of inertia into the rotational form of Newton's second law, and use the relation between linear and angular acceleration to derive the expression for α: α = (g/R)(m1−m2) / (m1+m2+M/2).
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