You are given a simple harmonic oscillator which is a mass o

You are given a simple harmonic oscillator, which is a mass on the end of a spring. The spring constant k=2 N/m and the mass M=0.5 kg. It is stretched to length of 0.2 meters and released with a velocity of -0.1 m/s (it is moving towards the equilibrium point) What is the angular frequency of oscillation? What is the period? what is the phase angle phi? initially, what is the potential energy, what is the amplitude? Find the maximum velocity at what time does the magnitude of the velocity reach its maximum (in seconds)? A rod with mass 1 kg. 0.5 m long is fastened down at a point 0.2 m from one end as in the picture. Two balls approach the rod both with velocity 0.1 m/s as shown and stick to the rod at the ends after collision. The first ball weighs 0.5 kg, and the second weights 0.6 kg. What is the final angular velocity of the system about the axis of rotation? Is the kinetic+potential energy conserved. If not. how much is lost or gained, (make sure to say if it is lost or gained...)

Solution

(a) angular frequency w= sqrt(k/m) here k is spring constant and m is the mass

w=sqrt( 2/0.5)=sqrt(4)=2 rad/sec

(b) we know f=(w/2pi)= 2/2*3.14=1/3.14

T=1/f =3.14 sec

(d) the potential energy energy= ( 1/2)k x² = (1/2)*2*0.04=0.04 joule

(e) the amplitude is = w²x=4*0.02=0.08m

(f) the maximum velocity= A w here A is amplitude and w is angular velocity =0.08*2=0.16 m/sec