Answer to Question #280834 in Mechanics | Relativity for john

Question

A plastic ball attached to a spring moves in simple harmonic motion. The amplitude of the ball's motion is 10.5 cm, and thi

spring constant is 7.50 N/m. When the ball is halfway between its equilibrium position and its maximum displacement fron

equilibrium, its speed is 27.0 cm/s.

(a) What is the mass of the ball (in kg)?

(b) What is the period of oscillation (in s)?

(c) What is the maximum acceleration of the ball? (Enter the magnitude in m/s?

Answer 1

(a) We can find the mass of the ball from the law of conservation of energy:

$$KE_i+PE_i=KE_f+PE_f,$$ $$\dfrac{1}{2}mv^2+\dfrac{1}{2}kx^2=0+\dfrac{1}{2}kA^2,$$ $$m=\dfrac{k(A^2-x^2)}{v^2},$$ $$m=\dfrac{7.5\ \dfrac{N}{m}\times((0.105\ m)^2-(0.0525\ m)^2)}{(0.27\ \dfrac{m}{s})^2}=0.85\ kg.$$

(b) We can find the period of oscillation as follows:

$$T=2\pi\sqrt{\dfrac{m}{k}}=2\pi\sqrt{\dfrac{0.85\ kg}{7.5\ \dfrac{N}{m}}}=2.11\ s.$$

(c) We can find the maximum acceleration of the ball as follows:

$$a_{max}=A\omega^2=A\dfrac{k}{m}=\dfrac{0.105\ m\times7.5\ \dfrac{N}{m}}{0.85\ kg}=0.93\ \dfrac{m}{s^2}.$$