Forces acting on a particle have magnitudes of 14, 7, and 7 N and act in the direction of vectors \(6\hat{i} + 2\hat{j} + 3\hat{k}\), \(3\hat{i} - 2\hat{j} + 6\hat{k}\), \(2\hat{i} - 3\hat{j} - 6\hat{k}\) respectively. The forces remain constant while the particle is displaced from point A: (2, –1, –3) to B: (5, –1, 1). The coordinates are specified in meters. The work done equal to:

1. 75 J 

2. 55 J 

3. 85 J 

4. 65 J

A body of mass m dropped from a height h reaches the ground with a speed of 1.4$\sqrt{\mathrm{gh}}$. The work done by air drag is:

1. –0.2mgh 

2. –0.02mgh

3. –0.04mgh 

4. mgh

A chain of length L and mass m is placed upon a smooth surface. The length of BA is (L–b). What will be the velocity of the chain when its end A reaches B?
        

1. \( \sqrt{\frac{2 g \sin \theta}{L}\left(L^2-b^2\right)} \)
2. \( \sqrt{\frac{g \sin \theta}{2 L}\left(L^2-b^2\right)} \)
3. \( \sqrt{\frac{g \sin \theta}{L}\left(L^2-b^2\right)}\)
4. None of these

A flexible smooth track is fixed in two alternate arrangements, as shown in figures 1 and 2. The length of the track used is the same in each case, and the height through which it falls from the bench to the floor is the same. A toy car is released at rest and slides down the track (One after the other on both the tracks). Air resistance can be ignored. Which of the following statement is true?
            
1. The speed at the bottom, as well as the time taken on both the tracks, are the same.

2. The speed at the bottom, as well as the time taken on both the tracks, are different.

3. The speed at the bottom is different but the time taken on both the tracks is the same.

4. The speed at the bottom is the same but the time taken on both the tracks is different.

A body of mass 2 kg moving with a velocity of $(\hat{\mathrm{i}}+$ $2\hat{\mathrm{j}}-$ $3\hat{\mathrm{k}})$ ms${}^{-1}$ collides with another body of mass 3 kg moving with a velocity of $(2\hat{\mathrm{i}}+$ $\hat{\mathrm{j}}+$ $\hat{\mathrm{k}})$ ms${}^{-1}$ . If they stick together, the velocity in ms${}^{-1}$ of the composite body will be:

1. $\frac{1}{5}(8\hat{i}+$ $7\hat{j}-$ $3\hat{k})$

2. $\frac{1}{5}(-4\hat{i}+$ $\hat{j}-$ $3\hat{k})$

3. $\frac{1}{5}(8\hat{i}+\hspace{0.17em}\hat{\mathrm{j}}-$ $\hat{\mathrm{k}})$

4. $\frac{1}{5}(-4\hat{i}+$ $7\hat{j}-$ $3\hat{k})$

If a 50 kg mass is swinging in a vertical plane on a string at rest then the power delivered by gravity when the mass is moving with a velocity of 2 m/sec upwards in a direction, making an angle of $60°$ with the vertical will be: (g = 9.8 m/$s^2$)

A force of 5 N making an angle $\theta$ with the horizontal acting on an object displaces it by 0.4 m along the horizontal direction. If the object gains kinetic energy of 1 J then the component of the force is:

When an object is shot from the bottom of a long, smooth inclined plane kept at an angle of \(60^\circ\) $$with horizontal, it can travel a distance \(x_1\) along the plane. But when the inclination is decreased to \(30^\circ\) $$and the same object is shot with the same velocity, it can travel \(x_2\) distance. Then \(x_1:x_2\) will be:
1. \(1:2\sqrt{3}\)
2. \(1:\sqrt{2}\)
3. \(\sqrt{2}:1\)
4. \(1:\sqrt{3}\)

A vertical spring with a force constant \(k\) is fixed on a table. A ball of mass \(m\) at a height \(h\) above the free upper end of the spring falls vertically on the spring so that the spring is compressed by a distance \(d\). The net work done in the process is:
1. \(mg(h+d)+\frac{1}{2}kd^2\)
2. \(mg(h+d)-\frac{1}{2}kd^2\)
3. \(mg(h-d)-\frac{1}{2}kd^2\)
4. \(mg(h-d)+\frac{1}{2}kd^2\)