A massless string is wrapped around a solid cylinder of mass 0.400 kg and radius 0.100 m. The string is attached to the ceiling. The cylinder is released. As it falls, the string unwinds. What is the magnitude of the acceleration of the cylinder as it falls?
Question
A massless string is wrapped around a solid cylinder of mass 0.400 kg and radius 0.100 m. The string is attached to the ceiling. The cylinder is released. As it falls, the string unwinds. What is the magnitude of the acceleration of the cylinder as it falls?
Solution
To solve this problem, we need to use the principles of physics, specifically Newton's second law and the concept of rotational motion.
Step 1: Identify the forces acting on the cylinder. There are two forces acting on the cylinder: the force of gravity (Fg = m*g, where m is the mass of the cylinder and g is the acceleration due to gravity) and the tension in the string (T).
Step 2: Apply Newton's second law. The net force acting on the cylinder is equal to the mass of the cylinder times its acceleration (F_net = m*a). In this case, the net force is the difference between the force of gravity and the tension in the string (F_net = Fg - T).
Step 3: Consider the rotational motion of the cylinder. The tension in the string causes the cylinder to rotate as it falls. The torque (τ) caused by the tension is equal to the tension times the radius of the cylinder (τ = Tr). This torque results in an angular acceleration (α) of the cylinder, which can be related to the linear acceleration (a) by the equation a = αr.
Step 4: Apply Newton's second law for rotation. The net torque acting on the cylinder is equal to the moment of inertia of the cylinder times its angular acceleration (τ_net = Iα). The moment of inertia of a solid cylinder is (1/2)mr^2, so we can write τ_net = (1/2)mr^2α. But since α = a/r, we can substitute to get τ_net = (1/2)mr*a.
Step 5: Set the expressions for net force and net torque equal to each other and solve for the acceleration. We have F_net = τ_net, so ma = (1/2)mra. Solving for a gives a = 2*g/(1 + r), where r is the radius of the cylinder.
Step 6: Substitute the given values into the equation for acceleration. The mass of the cylinder is 0.400 kg, the radius is 0.100 m, and the acceleration due to gravity is approximately 9.8 m/s^2. Substituting these values gives a = 2*9.8/(1 + 0.100) = 17.82 m/s^2.
So, the magnitude of the acceleration of the cylinder as it falls is approximately 17.82 m/s^2.
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