Work And Energy

To find the final kinetic energy, we use the work-energy theorem, which states that the work done by the net force equals the change in kinetic energy: 1. Calculate the work done:
- The force is N, and the displacement is m.
- Work done (dot product): 2. Apply the work-energy theorem:
- Initial kinetic energy .
- Work done .
- Using the work-energy theorem:
The final kinetic energy is 31 J, which matches option (2). Thus, the correct answer is (2).

The force acting on the particle is , where , and the particle is displaced from (since it starts at rest, we assume the initial position is ) to . We need to find the work done by the force over this displacement. Work done by a variable force is given by: Substitute and , with limits from to : Integrate: The work done by the force is 8 J, which matches option (4). Thus, the correct answer is (4).

The work done is given by the equation: where is the force and is the displacement. The force is calculated from Newton’s second law: where and . Thus, The displacement vector is given as: Now, compute the dot product : Thus, the work done is .

Non-conservative forces, such as friction or air resistance, do work that depends on the path taken. Unlike conservative forces (e.g., gravity), the work done by non-conservative forces is not related to the displacement between the initial and final points but depends on the actual path traversed by the object. - The statement "The work done by non-conservative force in a closed path is zero" is false, as work done by non-conservative forces (like friction) can be non-zero even in a closed path. - The statement "The work done by non-conservative forces does not depend on the path" is also incorrect, as the work done depends entirely on the path. - The correct statement is "The work done by non-conservative forces depends on the path," which is true, as the energy dissipated (e.g., by friction) depends on the path taken. - The statement "There is no energy loss in case of non-conservative forces" is false because non-conservative forces (like friction) lead to energy loss (often in the form of heat). Thus, the correct answer is option (3), as the work done by non-conservative forces depends on the path.