Question

Which of the following physical quantities has the same units as that of impulse?
A) Momentum
B) Force
C) Torque
D) Couple

Answer 1

Hint: Write down the dimensions of given physical quantities, whose physical quantity has the same dimension of impulse. Those have the same unit of impulse.

Complete step by step answer:
 Impulse is a certain amount of force you apply for a certain amount of time to cause a change in momentum.
\[\begin{align}
  & \operatorname{Im}pulse=Force\times time \\
 & \vec{J}=\vec{F}\times \Delta t \\
 & \Delta P=F\times \Delta t \\
\end{align}\]
$\Delta P=$ Change in momentum
$F=$ Applied force
$\Delta t=$ Elapsed time
We can make a direct connection between how a force acts on an object over time and the motion of the object. Impulse is useful in the real world, forces are often not constant.
The S.I unit of impulse is $newton-\sec ond$.
$\begin{align}
  & J=F\times \Delta t \\
 & J=m\times a\times \Delta t=Kg\times \dfrac{m}{{{\sec }^{2}}}\times \sec =Kg\dfrac{m}{\sec } \\
\end{align}$
Dimension of impulse
 $\begin{align}
  & J=\left[ M \right]\left[ L{{T}^{-2}} \right]\left[ T \right] \\
 & J=\left[ ML{{T}^{-1}} \right] \\
\end{align}$
Force:- A force or push or pull upon an object resulting from the objects interaction with another object. Whenever there is an interaction between two objects, there is a force upon each of the objects.
$\begin{align}
  & Force=Mass\times Acceleration \\
 & F=ma=Kg\times \dfrac{m}{{{\sec }^{2}}} \\
\end{align}$
Dimension of force
$\begin{align}
  & F=\left[ M \right]\left[ L{{T}^{-2}} \right] \\
 & F=\left[ ML{{T}^{-2}} \right] \\
\end{align}$
The dimension of force is different from the dimension of impulse, so impulse does not have the same unit.
Torque:- Torque is a measure of how much a force acting on an object causes that object to rotate.
$\begin{align}
  & \vec{\tau }=\vec{r}\times \vec{F} \\
 & \vec{\tau }=rF\sin \theta \\
\end{align}$
$\vec{\tau }=r\times m\times a\times \sin \theta =m\times kg\times \dfrac{m}{{{\sec }^{2}}}=\dfrac{{{m}^{2}}kg}{{{\sec }^{2}}}$
Dimension of Torque
$\begin{align}
  & \tau =\dfrac{\left[ M \right]\left[ {{L}^{2}} \right]}{\left[ {{T}^{2}} \right]} \\
 & \tau =\left[ M{{L}^{2}}{{T}^{-2}} \right] \\
\end{align}$
Dimension of torque is different to the impulse, so impulse does not represent the Torque.
Momentum:- Momentum product of the mass of a particle and its velocity. Momentum is vector quantity.
Momentum $\vec{P}=m\vec{v}=kg\times \dfrac{m}{\sec }$
Dimension of momentum
$\begin{align}
  & P=\left[ M \right]\left[ L{{T}^{-1}} \right] \\
 & P=\left[ ML{{T}^{-1}} \right] \\
\end{align}$
So the dimension of momentum is the same as impulse, so momentum has the same unit of impulse.

Note: Two physical quantities can only be equated if they have the same units, here we can solve this question with a shortcut, as impulse is equal to change in momentum we can directly conclude that momentum has the same unit as that of impulse.