Set 9 Problem number 12

• Problem

• Solution

Problem

An ideal spring has restoring force constant 230 Newtons/meter. An unknown mass on the spring is observed to complete 76.92 cycles every minute. What is the unknown 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 76.92 cycles every minute.    This implies a frequency of 76.92 cycles/minute * 1 mimute / 60 sec = 1.282 cycles / sec.
• Since each cycle consists of 2 `pi radians, the object completes 2 `pi ( `freq) radians every second = 8.055 radians/second. This is the angular frequency.

Since we know k, we know that 8.055 radians/second = `sqrt[( 230 Newtons/meter) / m].

• Solving for m we obtain m = ( 230 Newtons/meter) / ( 8.055 radians/second) ^ 2 = 3.544 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.