Question

The potential energy of a particle of mass 5 kg moving in x-y plane is given by equation, U=-7x+24y joule. Here x and y are in meters at t=0, the particle is at origin and moving with velocity $(2\hat{i}+3\hat{j})$m/s. The magnitude of acceleration of particle is
A.$3m/{{s}^{2}}$
B.$5m/{{s}^{2}}$
C.$31m/{{s}^{2}}$
D.$15m/{{s}^{2}}$

Answer 1

Hint: To solve this problem we need to calculate the derivative of potential function with respect to x and y to calculate x and y components of force acting on 5 kg particle, further potential energy and conservative forces will be discussed and net force acting on the particle will be calculate by taking modulus of it.

Formula used:
Force
$\Rightarrow \vec{F}=-\dfrac{dU}{dx}-\dfrac{dU}{dy}$
Acceleration,
$\Rightarrow a=\dfrac{F}{m}$

Complete answer:
Potential energy for conservative forces can be defined as the amount of work done in moving a particle against conservative force to reach the final state of a particle, and these conservative forces do not depend on the path followed in the procedure; only the starting and ending point of motion are considered.
Now, the potential energy function is given by U=-7x+24y and mass of particle m= 5kg.
At t=0, velocity
$\Rightarrow \vec{v}=(2\hat{i}+3\hat{j})$
Now, force acting on the particle can be defined as the change in potential energy with respect to its position so,
$\Rightarrow \vec{F}=-\dfrac{dU}{dx}-\dfrac{dU}{dy}$
$\Rightarrow \vec{F}=-\dfrac{d(-7x+24y)}{dx}-\dfrac{d(-7x+24y)}{dy}$
$\Rightarrow \vec{F}=7\hat{i}-24\hat{j}$
$\Rightarrow {{\vec{F}}_{x}}=7$ and,
$\Rightarrow {{\vec{F}}_{y}}=-24$
Now, equivalent net force will be given by
$\Rightarrow \left| {\vec{F}} \right|=\sqrt{{{({{{\vec{F}}}_{x}})}^{2}}+{{({{{\vec{F}}}_{y}})}^{2}}}$
$\Rightarrow \left| {\vec{F}} \right|=\sqrt{{{7}^{2}}+{{(-24)}^{2}}}=\sqrt{625}=25N$
$\Rightarrow F=25N$
The acceleration of the particle is,
$\Rightarrow a=\dfrac{F}{m}$
$\Rightarrow a=\dfrac{25}{5}=5m/{{s}^{2}}$
$\therefore $ The magnitude of acceleration of particles in the x-y plane is $a=5m/{{s}^{2}}$,

Hence option (B) is correct.

Note:
When a body moves against conservative forces like gravitation or electric field etc. these forces are responsible for the negative work on the body and due to Newton’s third law body will exert positive work on them but instead of losing energy through work it will be stored in the form of potential energy of a system, for example if we lift a body it’s potential energy increases with the height because work has been done against the conservative force (gravitational force) and it will be stored in the form of potential energy.