Answer to Question #283570 in Physics for Fahad

Question

2. A metal block of mass M kg is attached to a spring of negligible mass and spring constant k as shown in Fig. 2, and is free to slide on a frictionless, horizontal surface. A clay ball of mass M/4 kg is fired at the block with velocity V m/s and sticks to it. The block is initially at rest and the spring is initially uncompressed.

(a) (3 marks) What is the speed of the block and ball system immediately after the impact?

(b) (4 marks) Write an equation for the position of the block as a function of time after the collision, assume that at t = 0 s, the instant of the impact, it is at X = 0 m, which is the unstretched point of the spring. Determine values for the amplitude, angular frequency in terms of the given quantities. Assume that X and V are positive to the left as shown.

(c) (3 marks) Calculate the time period, maximum velocity, and maximum acceleration of the motion of the spring block and ball system.

Answer 1

(a) Apply momentum conservation:

$$\dfrac M4v=\bigg(\dfrac M4+M\bigg)u,\\\space\\ u=\dfrac v5.$$

(b) Apply energy conservation:

$$\dfrac{1}{2}\dfrac{5m}{4}\bigg(\dfrac v5\bigg)^2=\dfrac12kA^2,\\\space\\ A=\dfrac v2\sqrt\dfrac{m}{5k}.$$

As we know:

$$u=\omega A→\omega=u/A,\\\space\\ \omega=2\sqrt\dfrac{k}{5m}.$$

Position-time equation:

$$x(t)=A\sin(\omega t)=\dfrac v2\sqrt\dfrac{m}{5k}·\sin\bigg(2\sqrt\dfrac{k}{5m}·t\bigg)$$

(c) Time period:

$$T=\dfrac{2\pi}\omega=\pi\sqrt\dfrac{5m}{k}.$$

Maximum velocity:

$$v_\text{max}=u=v/5.$$

Maximum acceleration:

$$a_\text{max}=\omega^2A=\dfrac{2kv}{5m}\sqrt\dfrac{m}{5k}=\dfrac{2v}{5}\sqrt\dfrac{k}{m}.$$