Question

A mass spring system moves with simple harmonic motion along the x axis between turning points at $x_1=50\text{cm}$, and $x_2=60\text{cm}$. For parts (i) through (iii), choose from the same five possibilities. (i) At which position does the particle have the greatest magnitude of momentum? (a) 20 cm (b) 30 cm (c) 40 cm (d) some other position (e) The greatest value occurs at multiple points. (ii) At which position does the particle have greatest kinetic energy? (iii) At which position does the particle-spring system have the greatest total energy?

Short answer

1. Option (c) is correct.
2. Option (c) is correct.
3. Option (e) is correct.

Step-by-step solution

Simple Harmonic motion

When an object undergoes simple harmonic motion, the position as a function of time may be written as

$x=A\cos \left(\omega t\right)$

$A=$Amplitude

$\omega =$Angular frequency

$x=$Position of object

Step 2(i): Find at which position does the particle have the greatest magnitude of momentum

At 40 cm we have the midpoint between the turning points, so it is the equilibrium position and the point of maximum speed, and therefore, maximum momentum. Option (c) is correct.

Step 3(ii): Find at which position does the particle have greatest kinetic energy 

The position of maximum speed is also the position of maximum kinetic energy.

Option (c) is correct.

Step 4(iii): Find at which position does the particle-spring system have the greatest total energy

The total energy of the system is conserved, so it is the same at every position.

Option (e) is correct.