Time Period of a Mass–Spring System (DP IB Physics)

A 200 g toy robot is attached to a pole by a spring, which has a spring constant of 90 N m⁻¹ and made to oscillate horizontally.

Calculate:

(a) The force that acts on the robot when the spring is extended by 5 cm.

(b) The acceleration of the robot whilst at its amplitude position. 

Answer:

(a)

• Consider the motion of the robot at the equilibrium and stretched (amplitude) positions:

• The restoring force is given by

• Using:

  • Extension,  = 5 cm = 0.05 m

  • Spring constant,  = 90 N m⁻¹

= −90 × 0.05 = −4.5 N

• A force of 4.5 N will act on the robot, trying to pull it back towards the equilibrium position. 

(b)

• Newton's second law relates force and acceleration by

• Using:

  • Mass, m = 200 g = 0.2 kg

=  = −22.5 m s⁻²

• The robot will decelerate at a rate of 22.5 m s⁻² when at this amplitude position

Calculate the frequency of a mass of 2.0 kg attached to a spring with a spring constant of 0.9 N m^–1 oscillating with simple harmonic motion.

Answer:

Step 1: Write down the known quantities

• Mass, m = 2.0 kg

• Spring constant, k = 0.9 N m⁻¹

Step 2: Write down the equation for the time period of a mass-spring system

Step 3: Combine with the equation relating time period T and frequency, f

Step 4: Substitute in the values to calculate frequency

Frequency:  f = 0.11 Hz

Another area of physics where you may have seen the spring constant k is from Hooke's Law. Exam questions commonly merge these two topics together, so make sure you're familiar with the Hooke's Law equation too.

In the second worked example, the frequency calculated is the natural frequency of the mass-spring system, a concept that comes up in the topic of resonance.