Question

The angularmomenta$\mathbf{l}\left(\mathbf{t}\right)$. of a particle in four situations are (1)$\text{l=3t+4}$(2)$\mathbf{}\mathbf{l}\mathbf{=}\mathbf{-}6{\mathbf{t}}^2$(3$\mathbf{l}\mathbf{}\mathbf{=}\mathbf{}\mathbf{2}$(4)$\mathbf{l}\mathbf{}\mathbf{=}\mathbf{}\mathbf{4}\mathbf{/}\mathbf{t}\mathbf{}$In which situation is the net torque on the particle (a) zero, (b) positive and constant, (c) negative and increasing in magnitude (t>0), and (d) negative and decreasing in magnitude (t>0)?

Short answer

(a) In the third situation net torque is zero

(b) In the first situation net torque is positive and constant

(c) In the second situation net torque is negative and increasing

(d) In the fourth situation net torque is negative and decreasing

Step-by-step solution

Step 1: Given data

Angular momenta are,

1. $l=3t+4,$
   
2. $l=-6t^2,$
   
3. $l=2,$
   
4. .$l=4/t$

Understanding the concept of angular momentum

Torque is the rate of change of angular momentum (l). By taking the derivative of l, we find the net torque for a given situation.

Formulae are as follows:

$\tau =\frac{dl}{dt}$

(a) Determining the situation when net torque is zero

Now, calculate for the first situation $l=3t+4$:

$\begin{array}{c}\tau =\frac{dl}{dt}\\ =\frac{d(3t+4)}{dt}\\ =3\end{array}$

So, the net torque is positive and constant.

(b) Determining the situation when net torque is positive and constant

Now, calculatefor the second situation$l=-6t^2$:

$\frac{d(-6t^2)}{dt}=-12t$

So, the net torque is negative and increasing.

(c) Determining the situation when net torque is negative and increasing

Now, calculatethe third situation$l=2$ :

$\tau =\frac{d\left(2\right)}{dt}=0$

So, the net torque is zero.

(d) Determining the situation when net torque is negative and decreasing

Now, calculate for the fourthsituation$l=4/t$:

.$\tau =\frac{d\left(\frac{4}{t}\right)}{dt}=-4t^{-2}$

So, the net torque is negative and decreasing magnitude.

Therefore, find the net torque by taking the time derivative of the given angular momenta.

Then determine whether the torque is positive, negative, constant increasing or decreasing, or zero.