Question

A block with mass m = 0.1 kg oscillates with amplitude A = 0.1 m at the end of a spring with force constant k = 10 N/m on a frictionless, horizontal surface. Rank the periods of the following situations from greatest to smallest. If any periods are equal, show their equality in your ranking. (a) The system is as described above. (b) The system is as described in situation (a) except the amplitude is 0.2 m. (c) The situation is as described in situation (a) except the mass is 0.2 kg. (d) The situation is as described in situation (a) except the spring has force constant 20 N/m. (e) a small resistive force makes the motion under damped.

Short answer

The ranking is $\left(c\right)>\left(e\right)>\left(a\right)=\left(b\right)>\left(d\right)$.

Step-by-step solution

Step 1: Position and initial velocity of the particle

$T=2\pi \sqrt{\frac{m}{k}}$

$T=$Tension in the string

$k=$Spring constant

$m=$Mass of object

Step 2: Show their equality in your ranking

• The ranking is$\left(c\right)>\left(e\right)>\left(a\right)=\left(b\right)>\left(d\right)$. The amplitude does not affect the period in simple harmonic motion; neither do constant forces that offset the equilibrium position. Thus (a) and (b) have equal periods.
• The period is proportional to the square root of mass divided by spring constant. So (c), with larger mass, has a larger period than (a).
• And (d) with greater stiffness has smaller period. In situation (e) the motion is not quite simple harmonic, but has slightly smaller angular frequency and so a slightly longer period.