A particle is released from height \(S\) from the surface of the Earth. At a certain height its kinetic energy is three times its potential energy. The height from the surface of earth and the speed of the particle at that instant are respectively
1. \(\frac{S}{4}\), \(\sqrt{\frac{3gS}{2}}\)
2. \(\frac{S}{4}\), \(\frac{3gS}{2}\)
3. \(\frac{S}{4}\), \(\frac{\sqrt{3gS}}{2}\)
4. \(\frac{S}{2}\), \(\frac{\sqrt{3gS}}{2}\)
View Answer
Total mechanical energy is \(E = mgS\). At height \(h\), \(KE = 3 PE⇒ E = KE + PE = 4 PE ⇒ mgS = 4mgh ⇒ h = S/4\). Also, \(\frac{1}{2}mv^2 = 3mg(S/4) ⇒ v = \sqrt{\frac{3gS}{2}}\).
Water falls from a height of \(60\text{ m}\) at the rate of \(15\text{ kg/s}\) to operate a turbine. The losses due to frictional force are \(10%\) of the input energy. How much power is generated by the turbine? (\(g = 10\text{ m/s}^2\))
1. 7.0 kW
2. 10.2 kW
3. 8.1 kW
4. 12.3 kW
View Answer
The total input power is \(P_{\text{in}} = \frac{dm}{dt} gh = 15 \times 10 \times 60 = 9000\text{ W} = 9\text{ kW}\). Given that \(10%\) of energy is lost to friction, the efficiency is \(90%\), so the power generated is \(P_{\text{out}} = 0.9 \times 9\text{ kW} = 8.1\text{ kW}\).
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
The energy stored in a spring is \(U = \frac{1}{2}kx^2\). If stretched by 1 cm, \(U_1 = 50\text{ J}\). If further stretched by 1 cm, total stretch becomes 2 cm, so the stored energy is \(U_2 = \frac{1}{2}k(2x)^2 = 4U_1 = 200\text{ J}\).
Assertion (A): The work done during a round trip is always zero.
Reason (R): No force is required to move a body in its round trip.
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
Work done by conservative forces in a closed loop is zero. For all forces, it depends on energy change and non-conservative forces. Force is required to move a body, especially to overcome inertia or friction. Therefore, both Assertion (A) and Reason (R) are false.
Assertion (A): The sum of potential and kinetic energy for a system of moving objects is conserved only when no net external force acts on the objects
Reason (R): If no nonconservative force acts on a system of objects, the work done by external forces on a system of objects is equal to change in potential energy plus change in kinetic energy of the system.
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
Assertion is false: Mechanical energy \(E = K + U\) is conserved when only conservative forces do work. External conservative forces (e.g., gravity) can act and do work, yet mechanical energy can be conserved.
Reason is false: If no non-conservative forces act, then \(\Delta K + \Delta U = 0\), meaning mechanical energy is conserved. In this scenario, the work done by external forces \(W_{ext}\) is not necessarily zero. For instance, gravity does work when an object falls, but \(E\) is conserved.
Therefore, both Assertion and Reason are false.
Assertion (A): An athlete accelerates from rest to its maximum speed due to friction between his shoes and track.
Reason (R): Positive work done by frictional force increases the kinetic energy of athlete.
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
Assertion is true: An athlete pushes backward on the ground, and the ground exerts a forward static friction force on the athlete, causing acceleration.
Reason is true: The static friction force acts in the direction of the athlete's motion. Therefore, it does positive work, which directly increases the athlete's kinetic energy \(K\) according to the work-energy theorem.
Both A and R are true, and R provides the correct explanation for A.
Assertion (A): Work done by conservative force along closed path is zero.
Reason (R): When an object is moved along closed path beginning and ending are at same point its displacement is zero.
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
Assertion is true by definition of a conservative force, which does zero work over any closed path.
Reason is also true as displacement is zero for a closed path, i.e., \(\Delta \vec{r} = 0\). However, the reason is not the correct explanation for why work by a conservative force is zero; this is due to path independence and \(\Delta U = 0\) for a closed path.
Thus, both are true, but R is not the correct explanation of A.