Question

1. A cylinder of mass m, height h and radius r floats partially submerged in a liquid of density ρ. One third of the height of the cylinder is above the surface of the water. Johnny pushes the cylinder down by a small distance x<h/3, then he releases it from rest.

A) prove that the resulting motion of the cylinder is a simple harmonic motion.

B) Find the period of the small oscillations in terms of the given quantities (m,r,h,ρ)

2. An object of mass m is suspended from a vertical spring of force constant 1800 N/m. When the object is pulled down 2.5 cm from its equilibrium position and released from rest, it oscillates at 5.5 Hz.

A)Find the mass of the object.

B)Considering t=0 the moment of release, write the expressions for the displacement, acceleration, velocity and net force as a function of time

*********PLEASE EXPLAIN IN GREAT DETAIL THANKS***************

Answer 1

2

A)

k = spring constant = 1800 N/m

m = mass of the object hanging = ?

f = frequency of oscillation = 5.5

Frequency of oscillation is given as

f = (1/(2$\pi$)) sqrt(k/m)

5.5 = (1/(2 x 3.14)) sqrt(1800/m)

m = 1.51 kg

B)

A = amplitude of motion = 2.5 cm = 0.025 m

w = angular frequency = 2 $\pi$f = 2 x 3.14 x 5.5 = 34.54 rad/s

displacement equation is given as

x(t) = A Coswt

x(t) = (0.025) Cos(34.54 t)

taking derivative both side relative to "t"

dx/dt = (0.025) (- Sin(34.54 t)) (34.54)

v(t) = - 0.86 Sin(34.54 t)

taking derivative relative to "t" both side

dv(t)/dt = (d//dt) (- 0.86 Sin(34.54 t))

a(t) = (- 0.86) (34.54) Cos(34.54 t)

a(t) = - 29.7 Cos(34.54 t)

net force is given as

F(t) = m a(t)

F(t) = (1.51) (- 29.7 Cos(34.54 t))

F(t) = - 44.85 Cos(34.54 t)