Question

1. Many tall building have mass dampers, which are anti-sway devices to prevent them from oscillating in a wind. The device might be a block oscillating at the end of a spring and on a lubricated track. If the building sways, say eastward, the block also moves eastward but delayed enough so that whern it finally moves, the building is then moving back westward Thus, the motion of the oscillator is out of step with the motion of the building. .Suppose that the block has mass and is designed to oscillate at frequency and with amplitude (a) What is the total mechanical energy of the spring-block system? (b) What is the blocks speed as it passes through the equilibrium point?

Answer 1

"What is the total mechanical energy of the spring-block system. The mechanical energy E (the sum of the K.E = 1/2 mv^2 of the block and the P.E = 1/2 kx^2 of the spring) is constant throughout the motion of the oscillator. They we can evaluate E at any point during the motion. Suppose amplitude x_m of the oscillations, let's evaluate E when the block is at position. x = x_m, where ""U"" at that point. We first need to find the spring constant K. w = squareroot k/m w = 2 pi f squaring eq^n -(1) w^2 = k/m k = mw^2 [w = 2 pi f] k = m(2 pi f)^2 T.E = K.E + P.E = 1/2 mv^2 + 1/2 kx^2 passes through the \r\n what is the block's speed as it passes through the equilibrium point. The speed at x = 0 where the P.E = 1/2 kx^2 = 1/2 k(0)^2 = 0 Total energy (T.E) = "