Answer to Question #128062 in Physics for Ash Croft

Question

Let the mass of the body is 25 gm, the force constant k be 400 dynes/cm, and let the motion be started
by displacing the body 10 cm to the right of its equilibrium position and imparting to it a velocity toward
the right of 40 cm/s. Compute (a) the period, T, (b) the frequency f, (c) the angular frequency ω, (d) the
total energy E, (e) the amplitude A, (f) the angle θo, (g) the maximum velocity vmax ,(h) the maximum
acceleration amax , (i) the coordinate equations, velocity, and acceleration, and (j) displacement, velocity,
acceleration at a time �/8 sec after the start of the motion

Answer 1

a)

$$T=2\pi\sqrt{\frac{0.025}{0.4}}=1.57 s$$

b)

$$f=\frac{1}{1.57}=0.637\ Hz$$

c)

$$\omega=\frac{0.637}{2\pi}=0.101\frac{rad}{s}$$

d)

$$E=0.5(0.025)(0.4)^2+0.5(0.4)(0.1)^2=0.004\ J$$

e)

$$0.5(0.4)A^2=0.004\to A=0.141\ m$$

f)

$$0.1=0.141\cos{\theta_0}\to 44.8\degree$$

g)

$$0.5(0.025)V^2=0.004\to V=0.566\frac{m}{s}$$

i)

$$x=0.141\cos{(0.782+0.101t)}\\v=0.566\sin{(0.782+0.101t)}\\a=-0.0572\cos{(0.782+0.101t)}$$