Question

Which of the following quantities are always zero in SHM?
A. $\vec{F} \times \vec{a}$
B. $\vec{v} \times \vec{r}$
C. $\vec{a} \times \vec{r}$
D. $\vec{F} \times \vec{r}$

Answer 1

Recall that a simple harmonic motion entails a to and fro motion of an oscillator about its mean or equilibrium position. Additionally, we know that a simple harmonic motion demands that the restoring force acts on the oscillator in a direction opposite to the displacement of the oscillator, and so does the acceleration. In such a case, determine the angle between the quantities listed in the option and consequently find their cross product as a function of the sine of the angle between them, which will consequently lead you to the appropriate result.

Formula used:
Condition for SHM: $\vec{F} \propto -\vec{r}$, and
$\vec{a} \propto -\vec{r}$

Complete step by step answer:
Let us begin by establishing an understanding of SHM or Simple Harmonic Motion.
SHM is a type of periodic motion where the force acting on the oscillating object is directly proportional to the magnitude of the object’s displacement and is always directed towards the object’s equilibrium or mean position. This is given as:
$\vec{F}_{restoring} \propto -\vec{r}$, where the negative sign indicates that the restoring force is directed towards the harmonic oscillator’s equilibrium position. The force acting on a body executing SHM is a restoring force acting along the harmonic path of the oscillating body that facilitates the periodic oscillations of the body.
Thus, the conditions for SHM are as follows:
$\vec{F} \propto -\vec{r}$, and
$\vec{a} \propto -\vec{r}$, where r is the displacement of the body, a is the acceleration of the body, and F is the restoring force acting linearly along the body’s line of motion.

Now, we know that the cross product between two vectors $\vec{a}$ and $\vec{b}$ is defined as:
$\vec{a} \times \vec{b} = |a||b|sin\theta$.
Thus, the vector (cross) product is defined based on the sine of the angle between any two given vectors.
In this case, from the conditions that we established above for SHM, we have,
$\vec{F}$ and $\vec{r}$ acting collinearly in opposite directions indicated by the negative sign. This means that the angle $\theta$ between them is $\theta = 180^{\circ} \Rightarrow sin(180^{\circ}) = 0$.
Similarly, $\vec{a}$ and $\vec{r}$ act collinearly in opposite directions as well, since $\vec{F}=m\vec{a} \Rightarrow \vec{a} \propto \vec{F} \propto -\vec{r} \Rightarrow \vec{a} \propto -\vec{r}$.
$\Rightarrow \vec{F} \times \vec{r} = |F||r|sin(180^{\circ}) = 0 = \vec{a}\times \vec{r}$
From the above analysis, we also established that $\vec{a} \propto \vec{F}$, which means that the acceleration of the body produced as a consequence of the restoring force acts along the same direction as the restoring force. This means that $\theta$ between $\vec{F}$ and $\vec{a}$ is $\theta = 0^{\circ} \Rightarrow sin(0^{\circ})=0$
$\Rightarrow \vec{F} \times \vec{a} = |F||a|sin(0^{\circ})=0$
Similarly, the velocity produced by the displacement of the body is in the direction of this displacement. Therefore, the angle between $\vec{v}$ and $\vec{r}$ is $\theta = 0^{\circ}$.
Therefore,
$\vec{v} \times \vec{r} = |v||r|sin(0^{\circ})=0$
Thus, from the options given to us we see that all the quantities possess a zero value in SHM.
The correct choice(s) would be A, B, C and D.

Note:
Remember that for a particle executing SHM, at mean position, the velocity of the particle is maximum and the acceleration of the particle is minimum, whereas at the extreme position, the velocity is minimum and the acceleration of the particle is maximum. The direction of acceleration is equivalent to that of the restoring force, whereas the direction of velocity is equivalent to the direction of displacement of the oscillator.