Question

Let’s think back to the lab for simple harmonic motion. Consider the setup for the simple pendulum. The length of the pendulum is 0.60 m and the bob has inertia 0.50 kg (assume mass of the string is negligible, and small angular displacements). You conduct two experiments (A and B) to investigate the physics of simple harmonic motion. For experiment A, you pull the pendulum 5 ◦ , or π 36 radians. In experiment B, the angular displacement is 10◦ , or π 18 radians. Define clockwise rotation as positive.

(e) What are the maximum kinetic energies and potential energies for both experiments?

(f) In general, where do these occur for the motion of a pendulum?

(g)How do the total energies compare for the two experiments?

(h) For experiment A, write the equation of motion for angular displacement, θ(t) =??? (use either a sine or cosine and remember that clockwise is positive). Let t = 0 be the instant the bob passes though equilibrium moving in the counter-clockwise direction.

Answer #1

=
5^{o}

=
10^{o}

m =0.5 kg, l= 0.6m

e) P.Emax = m*g*h_{max}

cos() = x/l => x = l*cos()

=> hmax = l-x =l - l*cos()

for case 1(5o ):

hmax = l-x =l - l*cos() = 0.6 -
0.6*cos(5^{o}) = 2.283*10^-3m

P.E_{max}=0.5*9.81*2.283*10^-3=0.011198J

According conservation of energy, at P.Emax is converted into K.Emax at h=0

K.Emax = 0.011198J

for case2(10^{o}):

hmax = l-x =l - l*cos() = 0.6 -
0.6*cos(10^{o}) =9.115*10^-3m

P.E_{max}=0.5*9.81*9.115*10^-3=0.0447J

K.Emax =0.0447J

f) P.Emax at extreme ends where h= hmax and K.E=0 (v=0)

and K.Emax at equilibrium position(h=0) where P.E= 0 and KE=max

g) E1/E2 = 0.2505

h)

= −
*g **sin *θ* ⁄ *l* = -9.81*0.0871/0.6
=-1.42499

θo(t=0) =0

θ(t) =1/2* (-1.42499 )*t^2

1)x = (9.2 m) cos[(5πrad/s)t + π/4
rad]
gives the simple harmonic motion of a body. At t = 2.1 s,
what are the (a) displacement,
(b) velocity, (c) acceleration,
and (d) phase of the motion? Also, what are the
(e) frequency and (f) period of
the motion?
2) An oscillating block-spring system takes 0.746 s to begin
repeating its motion. Find (a) the period,
(b) the frequency in hertz, and
(c) the angular frequency in radians per
second.

A mass oscillates on a horizontal spring in a simple harmonic
motion. It is observed that the duration of one full oscillation
cycle is 0.50 π seconds, and that the maximal displacement of the
oscillator from the equilibrium is 12.0 cm. Let time instant taken
as t = 0.00 sec correspond to the moment when the mass is at the
location 10.0 cm to the left of the equilibrium, and moving to the
left.
1. Find oscillator’s velocity and acceleration...

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