Simple Harmonic Motion Whiteboarding Exercises
1) An air-track cart attached to a spring completes one
oscillation every 2.4 s. At t = 0 the cart is released from
rest at a distance of 0.10 m from its equilibrium
position. What is the position of the cart at (a) 0.30 s,
(b) 0.60 s, (c) 2.7 s, and (d) 3.0 s?
Solution:
(a) Calculate x at time t = 0.30 s:
(b) Now use t = 0.60 s:
(c) Use t = 2.7 s:
(d) Use t = 3.0 s:
2) The air-track cart of the previous example is used for
this example. What are the velocity and acceleration of
the cart at (a) 0.30 s, (b) 0.60 s?
Solution:
Calculate the angular frequency:
(a) Calculate v at t = 0.30 s:
Calculate a at t = 0.30 s:
(b) Calculate v at t = 0.60 s:
Calculate a at t = 0.60 s:
3) On December 29, 1997, a United Airlines flight from
Tokyo to Honolulu was hit with severe turbulence 31
minutes after takeoff. Data from the airplane's "black
box" indicated the 747 moved up and down with an
amplitude of 30.0 m and a maximum acceleration of
1.8g. Treating the up-down motion of the plane as
simple harmonic, find (a) the time required for one
complete oscillation and (b) the plane's maximum
vertical speed.
Solution:
(a) Set the maximum acceleration equal to 1.8g.
Then solve for , and express in terms of T:
Solve for T, and substitute values:
(b) Calculate maximum speed:
4) A red delicious apple floats in a barrel of water. If you
lift the apple 2.0 cm above its floating level and release
it, it bobs up and down with a period of T = 0.750 s.
Assuming the motion is simple harmonic, find the
position, velocity, and acceleration of the apple at the
times (a) T/4 and (b) T/2.
Solution:
Find the amplitude:
Calculate the angular frequency:
(a) Find x:
Find v:
Find a:
(b) Find x:
Find v:
Find a:
5) A mass of 0.22 kg on an air-track cart is attached to a
spring, it oscillates with a period of 0.84 s. What is the
force constant for this spring?
Solution:
6) A 0.120 kg mass attached to a spring oscillates with an
amplitude of 0.0750 m and a maximum speed of 0.524
m/s. Find (a) the force constant and (b) the period of
motion.
Solution:
(a) Using maximum velocity find the angular velocity:
Then using equation (10) solve for k:
(b) Find the period:
7) When a 0.420 kg mass is attached to a spring, it
oscillates with a period of 0.350 s. If, instead, a
different mass,
, is attached to the same spring, it
oscillates with a period of 0.700 s. Find (a) the force
constant of the spring and (b) the mass
.
Solution:
(a) Find the spring constant using equation (12):
(b) Use equation (12) to find the new mass
:
8) A 0.260 kg mass is attached to a vertical spring. When
the mass is put into motion, its period is 1.12 s. How
much does the mass stretch the spring when it is at
rest in its equilibrium position?
Solution:
Find the spring constant using equation (12):
Then set the magnitude of the spring force equal to mg and solve for distance:
9) A 0.980 kg block slides on a frictionless, horizontal
surface with a speed of 1.32 m/s. The block encounters
an unstretched spring with a force constant of 245 N/m.
(a) How far is the spring compressed before the block
comes to rest? (b) How long is the block in contact
with the spring before it comes to rest?
Solution:
(a) Set the initial kinetic energy of the block equal to the spring
potential energy then solve for A:
(b) Calculate the period then divide by 4 (Why?):
The pendulum
take one second
2.00 seconds for
pendulum with a
10)
in a grandfather clock is designed to
to swing in each direction; that is,
a complete period. Find the length of a
period of 2.00 seconds.
Solution:
Use equation (14) and solve for L:
A pendulum is constructed from a string 0.627 m
long attached to a mass of 0.250 kg. When set in
motion, the pendulum completes one oscillation every
1.59 s. If the pendulum is held at rest and the string is
cut, how long will it take for the mass to fall through a
distance of 1.00 m?
11)
Solution:
Find the acceleration due to gravity:
Use the free-fall ideas to find the time the bob is in the air: