1. A mass of 2.5 kg is attached to a spring that has a value of k equal to 600 N m-1. (a) Determine the value of the damping constant c that is required to produce critical damping. (b) The mass receives an impulse that gives it a velocity of ? = 1 . 5 m/s at t = 0. Determine the maximum value of the resultant displacement and the time at which this occurs. 

2. A mass is attached to a horizontal spring (as shown in the Figure below) m has a value of 0.80 kg and the spring constant k is 180 N m-1. 

At time t = 0 the mass is observed to be 0.04 m further from the wall than the equilibrium position and is moving away from the wall with a velocity of 0.50 m s-1. Assume no damping effect. Obtain an expression for the displacement of the mass in the form x = A (cos ω t + φ ), obtaining numerical values for A , ω and φ . 

3. The figure shows three systems of a mass m suspended by light springs that all have the same spring constant k . Show that the frequencies of oscillation for the three systems are in the ratio of: ω a : ω b : ω c = √2 : 1 : √1 / 2.