Set 9 Problem number 12

• Problem

• Solution

Problem

A mass on a spring is observed to complete 181.8 cycles of oscillation every minute. The spring constant is known to be 700 Newtons/meter. What is the mass, in kilograms?

Solution

We know that the angular frequency of an object in simple harmonic motion is `sqrt(k/m). The angular frequency must be expressed in radians/second.

• The information given is that the object completes 181.8 cycles every minute.    This implies a frequency of 181.8 cycles/minute * 1 mimute / 60 sec = 3.03 cycles / sec.
• Since each cycle consists of 2 `pi radians, the object completes 2 `pi ( `freq) radians every second = 19.04 radians/second. This is the angular frequency.

Since we know k, we know that 19.04 radians/second = `sqrt[( 700 Newtons/meter) / m].

• Solving for m we obtain m = ( 700 Newtons/meter) / ( 19.04 radians/second) ^ 2 = 1.93 kilograms.

To solve symbolically, we solve `omega = `sqrt(k/m) for m, obtaining m = k / `omega ^ 2, then substitute the known values of k and the `omega found above.