A block of mass M hangs from a rubber cord. The block is supported so that the cord is not stretched. The unstretched length of the cord is L_0 and its mass is m, much less than M. The "spring constant" for the cord is k. The block is released and stops at the lowest point. (Use L_0 for L0, M, g, and k as necessary.)
(a) Determine the tension in the cord when the block is at this lowest point.
(b) What is the length of the cord in this "stretched" position?
(c) Find the speed of a transverse wave in the cord, if the block is held in this lowest position.
To solve this problem, we can use the principles of conservation of energy and Hooke's law. Let's start by answering each part of the question step by step:
(a) To find the tension in the cord when the block is at the lowest point, we need to consider the forces acting on the block. At the lowest point, the block is stationary, so the net force on it is zero. The forces acting on the block are the gravitational force and the tension in the cord. Since the cord is not stretched, the tension in the cord is equal to the weight of the block. The weight of the block is given by W = Mg, where M represents the mass of the block and g is the acceleration due to gravity. Therefore, the tension in the cord is T = Mg.
(b) To find the length of the cord in the stretched position, we need to consider how much the cord stretches when the block is hanging. The cord behaves as a spring, so we can use Hooke's law to describe its behavior. Hooke's law states that the force exerted by a spring is directly proportional to the amount the spring is stretched or compressed. Mathematically, it can be written as F = kx, where F is the force applied by the spring, k is the spring constant, and x is the displacement from the equilibrium position. In this case, the force applied by the cord is equal to the weight of the block (Mg) and the displacement of the cord is the stretched length (L - L0), where L is the total length of the cord. Therefore, we have Mg = k(L - L0). Solving for L, we get L = (Mg/k) + L0.
(c) To find the speed of a transverse wave in the cord when the block is held in the lowest position, we can use the equation v = sqrt(T/μ), where v is the speed of the wave, T is the tension in the cord, and μ is the linear mass density of the cord. The linear mass density is given by μ = (m/L0), where m is the mass of the cord and L0 is the unstretched length of the cord. Substituting the values, we get v = sqrt(T/(m/L0)). Using the tension found in part (a), we have v = sqrt((Mg)/(m/L0)).
So, in summary:
(a) The tension in the cord when the block is at the lowest point is T = Mg.
(b) The length of the cord in the stretched position is L = (Mg/k) + L0.
(c) The speed of a transverse wave in the cord when the block is held in the lowest position is v = sqrt((Mg)/(m/L0)).