Question

Average torque on a projectile of mass m, initial speed u and angle of projection θ between initial and final positions P and Q as shown in the figure about the point of projection is:
A. \[\dfrac{{m{u^2}\sin 2\theta }}{2}\]
B. \[m{u^2}\cos \theta \]
C. \[{u^2}\sin \theta \]
D. \[\dfrac{{m{u^2}\cos \theta }}{2}\]

Answer 1

Hint: We should know the definition of torque, angular momentum etc.
We should know the relationship between angular momentum and torque.

Complete step by step answer:
Average Torque would just be the time, average of that varying quantity averaged over the time it takes to complete a revolution. The basic equation between torque and angular momentum is, $\tau .\Delta {\text{t = }}\Delta {\text{L}}$

We know that the basic equation between torque and angular momentum is,

$\tau .\Delta {\text{t = }}\Delta {\text{L}}$

And angular momentum is the quantity of rotation of a body, which is the product of its moment of inertia and its angular velocity.

Torque is a force that caused machinery, etc. to turn \[\left( {rotate} \right)\]

We know from basic equation that,

$\tau .\Delta {\text{t = }}\Delta {\text{L}}$

$
  \Delta t{\text{ = }}\dfrac{{2u\sin \theta }}{g}\left[ {u = speed,\theta = angle\;{\text{of projection}}} \right] \\
  \Delta L = \left[ {{L_f} - {L_i}} \right]\;\left[ {{L_f} = find\;{\text{momentum}}} \right]\;\left[ {{L_i} = initial\;{\text{momentum}}} \right] \\
 $

Initial momentum is zero as it starts form a static point thus,

$
  \because \Delta L = \left( {{L_f}} \right) = m.u\sin \theta \times R \\
  mv\sin \theta \times \dfrac{{{u^2}\sin 2\theta }}{g} \\
  \dfrac{{m{u^3}\sin \theta .\sin 2\theta }}{g} \\
 $
Therefore we know that,
$
  {\tau _{ag}} = \dfrac{{\Delta L}}{{\Delta t}} \\
  \dfrac{{m{u^3}\sin \theta .\sin 2\theta }}{{g.\dfrac{{2u\sin \theta }}{g}}} = \dfrac{{m{u^2}\sin 2\theta }}{2} \\
 $

So, the correct answer is “Option A”.

Note:
We should take care of substitutions of different equations in one.
We should remember all basic equations.
We should make sure to avoid substitution errors.