Work Power And Energy

In this problem, we need to calculate the total work done by all forces on a particle moving on a straight line, given its velocity changes with distance according to the equation . The particle moves from to , and we aim to find the work done during this displacement.

The total work done on the particle by all forces can be found using the work-energy principle, which states that the work done on a particle equals its change in kinetic energy.

The kinetic energy of the particle at any position is given by:

Substituting into the kinetic energy formula, we get:

The work done, , is the change in kinetic energy as the particle moves from to :

At , the velocity , so .

At , the kinetic energy .

Hence, the work done is:

Thus, the total work done by all the forces applied on the particle during its displacement from to is:

Correct Answer:

This matches the provided correct answer option.

To determine the work done on the block, we need to consider the forces acting on it and the displacement. The force is applied at an angle of to the horizontal, and there is friction between the block and the surface.

1. The force can be divided into two components:
   • Horizontal component:
   • Vertical component:
2. Only the horizontal component does work on the block, as work is calculated along the direction of displacement.
3. The maximum frictional force is given by: where:
   • (coefficient of friction)
   • is the normal force
   • (acceleration due to gravity)
4. Let's calculate the work done:
5. Substitute the values:
6. Frictional Force:
7. The work done against friction is given by: where is the displacement.
8. Assuming equals the force of friction (slipping condition):
9. Thus the work done against friction:

Therefore, the correct answer is 245 J.

To determine which force cannot be expressed in terms of potential energy, we need to consider the characteristics of each listed force:

1. Coulomb's Force: This is the electrostatic force between two charged particles. It can be expressed in terms of potential energy because it is a conservative force. The potential energy associated with Coulomb's force can be written as where is Coulomb’s constant, and are the charges, and is the distance between the charges.
2. Gravitational Force: This force is exerted by masses due to gravity. It is also a conservative force and can be expressed in terms of potential energy, given by where is the gravitational constant, and are the masses, and is the distance between their centers.
3. Restoring Force: This is the force that brings a system back to equilibrium, such as in springs (Hooke's law). It is conservative, and its potential energy is expressed as where is the spring constant and is the displacement.
4. Frictional Force: This force opposes motion and is non-conservative. Because of its dissipative nature (it converts mechanical energy into heat), it cannot be expressed as potential energy. Instead, it is derived from the velocity of the moving object and acts to restrict motion, making it non-reversible and energy-dissipative.

Based on this analysis, frictional force is the force that cannot be expressed in terms of potential energy.

Conclusion: The frictional force is a non-conservative force and is the correct answer.