Assertion (A): The kinetic energy of a particle continuously increases with time if the resultant force on the particle must be at an angle less than \(90^{\circ}\) to the velocity at all instants.
Reason (R): The work done by the external forces on a system equals to change in total energy.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
View Answer
Assertion (A) is true.
Kinetic energy \(K\) increases if the net work done \(W_{\text{net}}\) is positive. Since \(W_{\text{net}} = \int \vec{F} \cdot d\vec{s}\) and \(d\vec{s}\) is in the direction of \(vec{v}\) , \(W_{\text{net}} > 0\) implies \(vec{F} \cdot \vec{v} > 0\). This means the angle between \(vec{F}\) and \(vec{v}\) must be less than \(90^{circ}\).
Reason (R) is also true, interpreting "total energy" as kinetic energy in the context of \(W_{\text{net}} = \Delta K\) for a particle or total mechanical energy for a system with external work. Reason (R) provides the fundamental principle behind Assertion (A). Therefore, both are true and R explains A.
Assertion (A): Kinetic energy of a system can be increased without applying any external force on the system.
Reason (R): If external forces are absent then work done by internal forces is equal to change in kinetic energy.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
According to the work-energy theorem, `\(W_{net} = \Delta KE\)`. If external forces are absent, the net work done on the system is only due to internal forces, i.e., `\(W_{int} = \Delta KE\)`. Thus, internal forces can increase kinetic energy, for example, in an explosion. Both assertion and reason are true, and the reason correctly explains the assertion.
Assertion (A): If a spring is compressed, energy is stored in spring and when it is elongated, energy is released.
Reason (R): Work done by spring force is equal to change in potential energy of the spring.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Elastic potential energy `\(U = \frac{1}{2} k x^2\)` is stored in a spring when it is compressed or elongated. Energy is released when the spring moves towards its equilibrium position, not during elongation itself. Thus, assertion (A) is false. For a conservative force like spring force, `\(W = -\Delta U\)`, so work done by spring force is equal to the negative of the change in potential energy.
Thus, reason (R) is also false. Both assertion and reason are false.
Assertion (A): Frictional forces are conservative forces.
Reason (R): Potential energy can be associated with frictional forces.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Frictional forces are non-conservative forces because the work done by them depends on the path taken and energy is dissipated as heat. Potential energy can only be associated with conservative forces (e.g., gravitational, elastic).
Therefore, both assertion and reason are false.
When a spring is stretched by 1 cm, it stores energy 50 J. If it is further stretched by 1 cm, the stored energy will be
1. 50 J
2. 100 J
3. 150 J
4. 200 J
View Answer
Energy is \(U = \frac{1}{2} k x^2\). Since \(U \propto x^2\), doubling the stretch from \(1\text{ cm}\) to \(2\text{ cm}\) increases the stored energy by a factor of \(2^2 = 4\). Thus, \(U' = 4 \times 50\text{ J} = 200\text{ J}\).