Work, Energy and Power

A particle is released from height S above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively.

1. S4, 3gS2
2. S2, 3gS2
3. S4, 3gS2
4. S2, 3gS2

Two cylindrical vessels of equal cross sectional area of 2 m2 contain water up to heights 10 m and 6 m, respectively. If the vessels are connected at their bottom then the work done by the force of gravity is (Density of water is 103 kg/m3 and g = 10 m/s2)

1. 1 105 J
2. 4 104 J
3. 8 104 J
4. 6 104 J

[figure] Consider two blocks A and B of masses m1 = 10 kg and m2 = 5 kg that are placed on a frictionless table. The block A moves with a constant speed v = 3 m/s towards the block B kept at rest. A spring with spring constant k = 3000 N/m is attached with the block B as shown in the figure. After the collision, suppose that the blocks A and B, along with the spring in constant compression state, move together, then the compression in the spring is, (Neglect the mass of the spring)

1. 0.3 m
2. 0.1 m
3. 0.4 m
4. 0.2 m

A block of mass 1 kg, moving along x with speed vi = 10 m/s enters a rough region ranging from x = 0.1 m to x = 1.9 m. The retarding force acting on the block in this range is Fr = -kx, with k = 10 N/m. Then the final speed of the block as it crosses the rough region is:

1. 4 m/s
2. 10 m/s
3. 8 m/s
4. 6 m/s

A body of mass ‘m’ connected to a massless and unstretchable string goes in vertical circle of radius ‘R’ under gravity g. The other end of the string is fixed at the center of circle. If velocity at top of circular path is n gR, where, n 1, then ratio of kinetic energy of the body at bottom to that at top of the circle is :

1. n2+4n2
2. n+4n
3. n2n2+4
4. nn+4

As shown below, bob A of a pendulum having massless string of length 'R' is released from 60 to the vertical. It hits another bob B of half the mass that is at rest on a frictionless table in the center. Assuming elastic collision, the magnitude of the velocity of bob A after the collision will be (take g as acceleration due to gravity.) [figure]

1. 43 Rg
2. 13 Rg
3. Rg
4. 23 Rg

A sand dropper drops sand of mass m(t) on a conveyer belt at a rate proportional to the square root of speed (v) of the belt, i.e., dmdt v. If P is the power delivered to run the belt at constant speed then which of the following relationship is true?

1. P v
2. P v
3. P2 v3
4. P2 v5

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). In the light of the above statements, choose the correct answer from the options given below:

1. Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
2. Both (A) and (R) are true and (R) is the correct explanation of (A)
3. (A) is false but (R) is true
4. (A) is true but (R) is false

A bead of mass 'm' slides without friction on the wall of a vertical circular hoop of radius 'R' as shown in figure. The bead moves under the combined action of gravity and a massless spring (k) attached to the bottom of the hoop. The equilibrium length of the spring is 'R'. If the bead is released from top of the hoop with (negligible) zero initial speed, velocity of bead, when the length of spring becomes 'R', would be (spring constant is 'k', g is acceleration due to gravity) [figure]

1. 2Rg + kR2m
2. 3Rg + kR2m
3. 2Rg + 4kR2m
4. 2 gR + kR2m

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R In the light of the above statements, choose the most appropriate answer from the options given below

1. A is false but R is true
2. Both A and R are true but R is NOT the correct explanation of A
3. A is true but R is false
4. Both A and R are true and R is the correct explanation of A