Question

The angular momentum of a particle about origin is varying as L = 4t + 8 (SI units) when it moves along a straight line y = x - 4 (x, y in meters). The magnitude of force acting on the particle would be:
A. 1 N
B. 2 N
C. $\sqrt{2}$ N
D. $\sqrt{3}$ N

Answer 1

Hint: We are given the angular momentum L and the rate of change of angular momentum should give us torque. Find the way of converting the given direction of its motion into a vector then use the torque formula to find the force.

Formula used:
The rate of change of angular momentum is torque:
$\tau ={\displaystyle \frac{dL}{dt}}$.
Torque acting on a body is also given by relation:
$\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}$.

Complete answer:
For the given particle, we have the angular momentum L = 4t + 8. This means that it is changing with time because as the value of t changes, L changes too. Therefore, the rate of change of angular momentum becomes:
$\tau ={\displaystyle \frac{dL}{dt}}={\displaystyle \frac{d(4t+8)}{dt}}=4$.
This means that the magnitude of torque is 4 units.

Now, the particle happens to be moving along a straight line y = x -4. This means that the position of the particle at any point is given by this equation. So, we can find the vector from the origin in the following manner:
1. Finding any two points that satisfy the equation for this line along which the particle moves. If we keep x = 0 we get y = - 4. If we keep x = 2, we get y = -2.
2. Now, our required vector will be joining these two points so, the vector going from (0, -4) to (2, -2) can be written as:
$\overrightarrow{r}=(2-0)\hat{i}+(-2-(-4))\hat{j}=2\hat{i}+2\hat{j}$.
The magnitude of this vector can be written as:
$r=\sqrt{2^2+2^2}=2\sqrt{2}$.

The magnitude of torque can be written simply as:
$\tau =rF$,
where F is the magnitude of force acting and r is the perpendicular distance from the axis.
So, we may write:
$F={\displaystyle \frac{\tau }{r}}={\displaystyle \frac{4}{2\sqrt{2}}}=\sqrt{2}$N.

So, the correct answer is “Option C”.

Note:
The tricky part in the question is to find out the vector $vecr$, which is supposed to be the distance perpendicular to which the force is acting. Also, as the question says 'angular momentum is varying as L = 4t + 8', one might assume that this is the rate of change of angular momentum but clearly as it is written in terms of L it is the expression for angular momentum itself.