Question

A time, dependent force $ F = 6t $ acts on a particle of mass $ 1kg $ .If the particles start work from rest, the work done by the force during the first second will be:
(A) $ 18J $
(B) $ 4.5J $
(C) $ 22J $
(D) $ 9J $

Answer 1

Hint: Since a time-dependent force is acting on the particle we can solve this problem using the concept of work-energy theorem.
According to the work-energy theorem the net work done by the forces acting on a particle equals change in kinetic energy.

Complete step by step solution:
According to work-energy theorem,
 $ W = \Delta K.E $
 $ \Rightarrow W = \dfrac{1}{2}m\left( {{v^2} - {u^2}} \right) $
We are given that,
 $ F = 6t $ $ m = 1kg $ $ u = 0 $ $ W = ? $
We know that
 $ F = ma = m\dfrac{{dv}}{{dt}} $
 $ \Rightarrow F = 1 \times \dfrac{{dv}}{{dt}} $
 $ {\text{Since f = 6t}} $
 $ \Rightarrow dv = 6tdt $
Now we are going to integrate
 $ \Rightarrow \int_0^v {dv = \int_0^1 {6tdt} } $
 $ \Rightarrow v = {\left[ {6\dfrac{{{t^2}}}{2}} \right]^1}_0 $
 $ \Rightarrow v = 3\left( {{1^2} - 0} \right) $
 $ \Rightarrow v = 3m{s^{ - 1}} $
 $ W = \Delta KE $
 $ \Rightarrow W = \dfrac{1}{2}m\left( {{v^2} - {u^2}} \right) $
 $ \Rightarrow W = \dfrac{1}{2} \times \left( {{3^2} - 0} \right) $
 $ \Rightarrow W = 4.5J $
Hence option B is the correct answer.

Additional Information:
Kinetic energy: It is the energy possessed by an object by virtue of its motion or movement. The SI unit of kinetic energy is Joule.
There are five different types of kinetic energy: Radiant energy, thermal energy, electrical energy and mechanical.
Radiant energy: It is the energy of electromagnetic waves which can travel through space. Light energy can be described as a form of radiant energy which is visible to the human eye.

Note:
Work energy theorem can be derived from Newton's second law. Work transfers energy from one place to another or one form to another. In general, systems can change the heat energy in a thermal system, the potential energy of a mechanical device, or the electrical energy in an electrical device. The principle of work and kinetic energy states that the work done by the sum of all the forces acting on a particle equals the change in the kinetic energy of the particle. This definition can be extended to rigid bodies by defining the work of the torque and rotational kinetic energy.