Question

A mass executes SHM at the end of a light spring. (a) What fraction of the total energy of the system is potential and what fraction is kinetic at the instant when the displacement of the mass is equal to half the amplitude? (b) If the maximum amplitude of the oscillations is doubled, what will be the change in (i) the total energy of the system, (ii) the maximum velocity of the mass and (iii) the maximum acceleration of the mass. Will the period of oscillation change?

Answer 1

Explained

Step-by-step explanation:

A) The total energy of the system is defined by the energy at maximum amplitude, which we'll call A. At that point, the energy of the system is

E = 1/2×m×A^2;

since energy is conserved, this is also the total amount of energy that the system ever has.

So at x=1/2A,

the potential energy of the system is 1/8×m×A^2

which is one-fourth of the system's total energy. Therefore, the remaining three-fourths is kinetic.

B) (i) Doubling the maximum amplitude will quadruple the total energy:

`E= (1)/(2)m(2A)^2`

(ii) Doubling the maximum amplitude will double the maximum velocity

`(1)/(2)m(2A)^2= (1)/(2)mV^2`

(iii) Doubling the maximum amplitude will double the maximum acceleration: m×a = -k(2A)

(iv) Doubling the maximum amplitude leaves the period unchanged:

`T= 2\pi\sqrt{(m)/(k) }`

(neither m nor k has changed).