Question

Problem 5. A 7.0kg bowling ball is hung to the ceiling via a 2.5 m steel wire. We move the ball horizontally, 10 cm away from the equilibrium position, and we start our chronometer when we let the ball go and it starts swinging. We model the system as an harmonic oscillator with elastic constant k expressed in terms of the mass of the ball m, the length 1 of the steel wire and g, the gravitational acceleration: k mg/l. Use these data for the following questions.1 1. Write the solution x(t) 2. Express the product of the energy times the period, E-T, in terms of the initial conditions x(0)-ao and p(O) 0. 3. How long does a full oscillation of the bowling ball take? 4. What is the energy E of the bowling ball? 5. Compute the ratio ET /h where ћ = 1.05 × 10-34 J . s is Planck's constant. A large value of the ratio ET/h confirms that quantum mechanical effects can be neglected in the oscillations of the bowling ball

Answer 1

5. given

m = 7 kg

l = 2.5 m steel wire

Ao = 10 cm

k = mg/l

1. the solution of such a simple harmonic motion with intiial conditions

xo = 10 cm

vo = 0 m/s

hence

x = 10 cm * cos(wt)

w = sqrt(g/l) = 1.9809 rad/s

2. now, at time t

v = -0.1wsin(wt)

a = -0.1w^2cos(wt)

PE = 0.5kx^2

KE = 0.5mv^2

TE = PE + KE = 0.5mv^2 + 0.5kx^2 = 0.5(k*xo^2sin^2(wt) + kxo*cos^2(wt)) = 0.5kxo^2

time periond T = 2*pi/w = 2*pi*sqrt(l/g)

hence

TE.T = 0.5*k*xo^2*2*pi*sqrt(l/g)

p = mv

po = mw*xo

po*xo = mw*xo^2

TE.T =pi* po*xo

3. T = 2*pi*sqrt(l/g) = 3.17186992 s

4. E = 0.5kxo^2 = 0.13734 J

5. ET/h = 0.414880571428*10^34