Question

A particle moves in the X-Y plane under the action of forces $F$ such that the values of linear momentum p at any time is ${p_x} = 2\cos t$ and ${p_y} = 2\sin t$. The angle between $\overrightarrow F $ and $\overrightarrow p $ at the time $t$ will be
A. ${0^o}$
B. ${30^o}$
C. ${90^o}$
D. ${180^o}$

Answer 1

Hint: We can solve this problem with Newton's laws of motion. There are three laws of motion. First laws of motion also known as law of inertia states that if a body is at rest position then it will remain at rest or if it is moving with some velocity it will be in moving position until the external force is not applied.

Complete answer:
Newton's second law states the rate of change of momentum of the body is directly proportional to the force applied. When the net force on the body is zero then the momentum of the body remains the same. So the net force is equal to the rate of change of momentum and change of momentum takes place in the direction of the force applied.

$F = \dfrac{{dp}}{{dt}} = \dfrac{{d\left( {mv} \right)}}{{dt}} = m\left( {\dfrac{{dv}}{{dt}}} \right) = ma$, where $m$ is the mass of the body, $a$ is the acceleration of body.
Force is known as rate of change of momentum, A particle moves in X-Y plane under the action of forces $F$ such that the values of linear momentum p at any times is ${p_x} = 2\cos t$ and ${p_y} = 2\sin t$, then
$
\overrightarrow p = 2\cos t\mathop i\limits^ \wedge + 2\sin t\mathop j\limits^ \wedge \\
F = \dfrac{{dp}}{{dt}} \\
 $ ,
${F_x} = - 2\sin t,{F_y} = 2\cos t$

From the dot product of the vector, $\overrightarrow A .\overrightarrow B = AB\cos \theta $ , where $\theta $ is the angle between these two vectors.

The angle between $\overrightarrow F $ and $\overrightarrow p $ at the time $t$ will be $\theta $ then from the dot product formula,

$\cos \theta = \dfrac{{\overrightarrow F .\overrightarrow p }}{{Fp}}$= $\dfrac{{ - 2\cos t \times 2\sin t + 2\cos t \times 2\sin t}}{{Fp}}$
$
  \cos \theta = 0 \\
  \theta = {90^0} \\
 $
The angle between $\overrightarrow F $ and $\overrightarrow p $ at the time $t$ will be $\theta $$ = {90^0}$.
Third law says that for every action there is an equal and opposite reaction.

So, the correct answer is “Option C”.

Note:
In simple words, the first law is: if the net force on the body is zero then the velocity remains the same and this motion is called uniform motion. Equation expressed,\[\sum\limits_{}^{} {F = 0 \Leftrightarrow \dfrac{{dV}}{{dt}} = 0} \]. The second law implies the conservation of momentum and the third law states that every action has its reaction force.