<< Chapter < Page Chapter >> Page >

The data in [link] can still be modeled with a periodic function, like a cosine function, but the function is shifted to the right. This shift is known as a phase shift    and is usually represented by the Greek letter phi ( ϕ ) . The equation of the position as a function of time for a block on a spring becomes

x ( t ) = A cos ( ω t + ϕ ) .

This is the generalized equation for SHM where t is the time measured in seconds, ω is the angular frequency with units of inverse seconds, A is the amplitude measured in meters or centimeters, and ϕ is the phase shift measured in radians ( [link] ). It should be noted that because sine and cosine functions differ only by a phase shift, this motion could be modeled using either the cosine or sine function.

Two graphs of an oscillating function of angle. In figure a, we see the function cosine of theta as a function of theta, from minus pi to two pi. The function oscillates between -1 and +1, and is at the maximum of +1 at theta equals zero. In figure b, we see the function cosine of quantity theta plus phi as a function of theta, from minus pi to two pi. The function oscillates between -1 and +1, and is maximum at theta equals phi. The curve is the cosine curve, shifted to the right by an amount phi.
(a) A cosine function. (b) A cosine function shifted to the right by an angle ϕ . The angle ϕ is known as the phase shift of the function.

The velocity of the mass on a spring, oscillating in SHM, can be found by taking the derivative of the position equation:

v ( t ) = d x d t = d d t ( A cos ( ω t + ϕ ) ) = A ω sin ( ω t + φ ) = v max sin ( ω t + ϕ ) .

Because the sine function oscillates between –1 and +1, the maximum velocity is the amplitude times the angular frequency, v max = A ω . The maximum velocity occurs at the equilibrium position ( x = 0 ) when the mass is moving toward x = + A . The maximum velocity in the negative direction is attained at the equilibrium position ( x = 0 ) when the mass is moving toward x = A and is equal to v max .

The acceleration of the mass on the spring can be found by taking the time derivative of the velocity:

a ( t ) = d v d t = d d t ( A ω sin ( ω t + ϕ ) ) = A ω 2 cos ( ω t + φ ) = a max cos ( ω t + ϕ ) .

The maximum acceleration is a max = A ω 2 . The maximum acceleration occurs at the position ( x = A ) , and the acceleration at the position ( x = A ) and is equal to a max .

Summary of equations of motion for shm

In summary, the oscillatory motion of a block on a spring can be modeled with the following equations of motion:

x ( t ) = A cos ( ω t + ϕ )
v ( t ) = v max sin ( ω t + ϕ )
a ( t ) = a max cos ( ω t + ϕ )
x max = A
v max = A ω
a max = A ω 2 .

Here, A is the amplitude of the motion, T is the period, ϕ is the phase shift, and ω = 2 π T = 2 π f is the angular frequency of the motion of the block.

Determining the equations of motion for a block and a spring

A 2.00-kg block is placed on a frictionless surface. A spring with a force constant of k = 32.00 N / m is attached to the block, and the opposite end of the spring is attached to the wall. The spring can be compressed or extended. The equilibrium position is marked as x = 0.00 m .

Work is done on the block, pulling it out to x = + 0.02 m . The block is released from rest and oscillates between x = + 0.02 m and x = −0.02 m . The period of the motion is 1.57 s. Determine the equations of motion.

Strategy

We first find the angular frequency. The phase shift is zero, ϕ = 0.00 rad, because the block is released from rest at x = A = + 0.02 m . Once the angular frequency is found, we can determine the maximum velocity and maximum acceleration.

Solution

The angular frequency can be found and used to find the maximum velocity and maximum acceleration:

ω = 2 π 1.57 s = 4.00 s −1 ; v max = A ω = 0.02 m ( 4.00 s −1 ) = 0.08 m/s; a max = A ω 2 = 0.02 m ( 4.00 s −1 ) 2 = 0.32 m/s 2 .

All that is left is to fill in the equations of motion:

x ( t ) = A cos ( ω t + ϕ ) = ( 0.02 m ) cos ( 4.00 s −1 t ) ; v ( t ) = v max sin ( ω t + ϕ ) = ( −0.08 m/s ) sin ( 4.00 s −1 t ) ; a ( t ) = a max cos ( ω t + ϕ ) = ( −0.32 m/s 2 ) cos ( 4.00 s −1 t ) .

Significance

The position, velocity, and acceleration can be found for any time. It is important to remember that when using these equations, your calculator must be in radians mode.

Got questions? Get instant answers now!

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 1' conversation and receive update notifications?

Ask