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1. The displacement \(x\) in meter of a particle of mass \(m\) kg moving in one direction under the action of a force is related to the time \(t\) in second by the equation \(x= (t-3)^2\). The work done by the force (in joules) in first six seconds is:
1. \(18m\)
2. Zero
3. \(9m\)
4. \(36m\)

2.

A mass M is lowered with the help of a string by a distance h at a constant acceleration g/2. The work done by the string will be
1. $\frac{Mgh}{2}$
2. $-\frac{Mgh}{2}$
3. $\frac{3Mgh}{2}$
4. $-\frac{3Mgh}{2}$

3.

A force of $\overrightarrow{F}=2x\hat{i}+2\hat{j}+3z^2\hat{k} N$ is acting on a particle. Find the work done by this force in displacing the body from (1, 2, 3) m to (3, 6, 1)m.
1. -10 J
2. 100 J
3. 10 J
4. 1 J

4.

If we shift a body in equilibrium from A to C gravitational field via path AC or ABC,
                       
1. The work done by the force $\overrightarrow{F}$ for both paths will be same
2. $W_{AC}>W_{ABC}$
3. $W_{AC}<W_{ABC}$
4. None of the above

5.

An engine pumps up 100 kg of water through a height of 10 m in 5 s. Given that the efficiency of the engine is 60%, what is the power of the engine? Take g= 10 ms^-2.
1. 3 kW
2. 3.3 kW
3. 0.33 kW
4. 0.033 kW

6.

An engine pumps water continuously through a hole. Speed with which water passes through the hole nozzle is v, and k is the mass per unit length of the water jet as it leaves the nozzle. Find the rate at which kinetic energy is being imparted to the water.
1. $\frac{1}{2} k{v}^2$
2. $\frac{1}{2} k{v}^3$
3. $\frac{v^2}{2k}$
4. $\frac{v^3}{2k}$

7.

A bus can be stopped by applying a retarding force F when it is moving with speed v on a level road. The distance covered by it before coming to rest is s. If the load of the bus increases by 50% because of passengers, for the same speed and same retarding force, the distance covered by the bus to come to rest shall be
1. 1.5s
2. 2s
3. 1s
4. 2.5s

8.

A heavy weight is suspended from a spring. A person raises the weight till the spring becomes slack. The work done by him is W. The energy stored in the stretched spring was E. What will be the gain in gravitational potential energy?
1. W
2. E
3. W + E
4. W - E

9.

The speed v reached by a car of mass m in traveling a distance x, driven with constant power P, is givenby
1. $v=\frac{3xP}{m}$
2. $v={\left(\frac{3xP}{m}\right)}^{1/2}$
3. $v={\left(\frac{3xP}{m}\right)}^{1/3}$
4. $v={\left(\frac{3xP}{m}\right)}^2$

10.

Figure shows the verticl section of a frictionless surface. A block of mass 2 kg is released from rest from position A; its KE as it reaches position C is (g= 10 m s^-2)
                                   
1. 140 J
2. 180 J
3. 120 J
4. 280 J

11.

The kinetic energy K of a particle moving along a circle of radius R depends upon the distance s as K=as². The force acting on the particle is 
1. $2a\frac{s^2}{R}$
2. $2as{\left[1+\frac{s^2}{R^2}\right]}^{1/2}$
3. 2as
4. 2a

12.

A block of 4 kg mass starts at rest and slides a distance d down a friction less incline (angle 30$°$) where it runs into a spring of negligible mass. The block slides an additional 25 cm before it is brought to rest momentarily by compressing the spring. The force constant of the spring is 400 Nm^-1. The value of d is (take g= 10 ms^-2)
                         
1. 25 cm
2. 37.5 cm
3. 62.5 cm
4. None of the above

13.

A particle is released one by one from the top of two inclined rough surfaces of height h each. The angles of inclination of the two planes are 30$°$ and 60$°$, respectively. All other factors (e.g., coefficient of friction, mass of block, etc.) are same in the both the cases. Let K₁ and K₂ be the kinetic energies of  the particle at the bottom of the plane in the two cases. Then
1. K₁ = K₂
2. K₁ > K₂
3. K₁ < K₂
4. Data insufficient

14.

The system shown in the figure is released from rest with mass 2 kg in contact with the ground. Pulley and spring  are massless, and  friction is absent everywhere. The speed of 5 kg block when 2 kg block leaves the contact with the ground is (found constant of the spring k= 40 Nm^-1 and g= 10 m s^-2)
               
1. $\sqrt{2} m s^{-1}$
2. $2\sqrt{2}m s^{-1}$
3. $2 m s^{-1}$
4. $\sqrt{2} m s^{-1}$

15.

A particle of mass m is projected at an angle $\alpha$ to the horizontal with an initial velocity u. The work done by the gravity during the time it reaches its highest point is
1. $u^2{\sin }^2\alpha$
2. $\frac{m{u}^2{\cos }^2\alpha }{2}$
3. $\frac{m{u}^2{\sin }^2\alpha }{2}$
4. $-\frac{m{u}^2{\sin }^2\alpha }{2}$

16.

A particle of mass m is projected at an angle $\alpha$ to the horizontal with an initial velocity u, the average power delivered by gravity is
1. $-mg u \cos \alpha$
2. $-mgu \sin \alpha$
3. $-\frac{mgu \cos \alpha }{2}$
4. $-\frac{mgu \sin \alpha }{2}$

17.

A person of mass 70 kg jumps from a stationary helicopter with the parachute open. As he falls through 50 m height, he gains a speed of 20 m s^-1.The work done by the viscous air drag is 
1. 21000 J
2. -21000 J
3. -14000 J
4. 14000 J

18.

A particle located in a one-dimensional potential field has its potential energy function as $U\left(x\right)=\frac{a}{x^4}-\frac{b}{x^2}$, where a and b are positive constants. The position of equilibrium x corresponds to 
1. $\frac{b}{2a}$
2. $\sqrt{\frac{2a}{b}}$
3. $\sqrt{\frac{2b}{a}}$
4. $\frac{a}{2b}$

19.

A collar B of mass 2 kg is constrained to move along a horizontal smooth and fixed circular track of radius 5 m. The spring lying in the plane of the circular track and having spring constant 200 N m^-1 is  underformed when the collar is at A. If the collar starts from rest at B, the normal reaction exerted by the track on the collar when it passes through A is 

1. 360 N
2. 720 N
3. 1440 N
4. 2880 N

20.

A particle of mass m slides on a frictionless surface ABCD, starting from rest as shown in figure. The part BCD is a circular arc. If it looses contact at point P, the maximum height attained by the particle from point C is
                                     
1. $R\left[2+\frac{1}{2\sqrt{2}}\right]$
2. $R\left[2+\frac{1}{2\sqrt{2}}\right]R$
3. 3R
4. None of these