Question

The length of a simple pendulum is 0.72 m and the mass of the particle (the "bob") at the end of the cable is 0.21 kg. The pendulum is pulled away from its equilibrium position by an angle of 8.40° and released from rest. Assume that friction can be neglected and that the resulting oscillatory motion is simple harmonic motion.

(a) What is the angular frequency of the motion?
____rad/s

(b) Using the position of the bob at its lowest point as the reference level, determine the total mechanical energy of the pendulum as it swings back and forth.
____ J

(c) What is the bob's speed as it passes through the lowest point of the swing?

____m/s

Answer 1

Solution: From the question we have:

The length of a simple pendulum is 0.72 m and the mass of the particle (the "bob") at the end of the cable is 0.21 kg.

The pendulum is pulled away from its equilibrium position by an angle of 8.40° and released from rest

Assume that friction can be neglected and that the resulting oscillatory motion is simple harmonic motion

Now,

SHM equ >> y = a sin wt = a sin (p)
find v (=dy/dt), then acceleration (f=dv/dt) you get
f = - w^2 y
---------
where w = 2pi f = 2pi/T is the angular frequency
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restoring force = - mg sin p
acceleration = - g sin p
now g sin p = w^2 y
here sin p = y/L (small angle)
g*y/L = w^2 y
---------------------------------
pendulum T = 2pi sqrt(L/g)
w=2pi/T = sqrt(g/L)
--------------------------
w^2 = g/L = 9.8/0.72
w = 3.68 rad/sec (answer 1)
---------------------------------
from above equ, one can show that
v^2 = w^2 (a^2 - y^2)
----------------------
KE of bob = 0.5 mv^2 = 0.5 m w^2 (a^2 - y^2)
PE of bob = 0.5 mw^2 * y^2
---------------------------------
PE = spring = 0.5 ky^2
k = m w^2
--------------------
ME = KE+PE = 0.5 mw^2 a^2
where a = amplitude = L sin p = 0.72* sin 8.4
ME = 0.5*0.21*(3.68)^2 * (0.72 sin 8.4)^2
ME = 0.012 Joule (answer2)
--------------------
speed at lowest point is maximum, and when y =0
v^2 = w^2 (a^2 - y^2)
v^2(mid) = (aw)^2
v(mid) = (aw) = 0.72 sin 8.4 * 3.68 = 0.348 m/s (answer 3)