Q. 12 Mass ${\mathrm{m}}_1$ moves on a slope making an angle $\mathrm{\theta }$ with the horizontal and is
attached to mass ${\mathrm{m}}_2$ by a string passing over a frictionless pulley as
shown in the figure. The coefficient of friction between ${\mathrm{m}}_1$ and the sloping
surface is $\mathrm{\mu }$. Which of the following statements are true?

$\begin{array}{l}\text{ (a) If }{\mathrm{m}}_2>{\mathrm{m}}_1\sin \mathrm{\theta }\text{ , the body will move up the plane }\\ \text{ (b) If }{\mathrm{m}}_2>{\mathrm{m}}_1(\sin \mathrm{\theta }+\mathrm{\mu cos}\mathrm{\theta }),\text{ the body will inove up the plane }\\ \text{ (c) If }{\mathrm{m}}_2<{\mathrm{m}}_1(\sin \mathrm{\theta }+\mathrm{\mu cos}\mathrm{\theta }),\text{ the body will move up the plane }\\ \text{ (d) If }{\mathrm{m}}_2<{\mathrm{m}}_1(\sin \mathrm{\theta }-\mathrm{\mu cos}\mathrm{\theta }),\text{ the body will move down the plane }\end{array}$

Q.15 A body of mass 10 kg is acted upon by two perpendicular forces, 6N and 8N. The resultant acceleration of the body is

$$\begin{gathered} \begin{array}{l} (\mathrm{a}) 1{\mathrm{ms}}^{-2} \mathrm{at} \mathrm{an} \mathrm{angle} \mathrm{of} {\tan }^{-1}\left(\frac{4}{3}\right) \mathrm{w} .\mathrm{r}.\mathrm{t}. 6\mathrm{N} \mathrm{force} \\ (\mathrm{b}) 0.2{\mathrm{ms}}^{-2} \mathrm{at} \mathrm{an} \mathrm{angle} \mathrm{of} {\tan }^{-1}\left(\frac{4}{3}\right) \mathrm{w}.\mathrm{r}.\mathrm{t}. 6\mathrm{N} \mathrm{force} \end{array} \\ \begin{array}{l} (\mathrm{c}) 1{\mathrm{ms}}^{-2} \mathrm{at} \mathrm{an} \mathrm{angle} \mathrm{of} {\tan }^{-1}\left(\frac{3}{4}\right) \mathrm{w}. .\mathrm{r}.\mathrm{t}. 8\mathrm{N} \mathrm{force} \\ (\mathrm{d}) 0.2{\mathrm{ms}}^{-2} \mathrm{a} \mathrm{t}\mathrm{an} \mathrm{angle} \mathrm{of} {\tan }^{-1}\left(\frac{3}{4}\right) \mathrm{w}.\mathrm{r}.\mathrm{t}. 8\mathrm{N} \mathrm{force} \end{array} \end{gathered}$$

A point mass 'm' is moved in a vertical circle of radius 'r' with the help of a string. The velocity of the mass is $\sqrt{7\mathrm{gr}}$ at the lowest point. The tension in the string at the lowest point is:

Calculate the acceleration of the block and trolly system shown in the figure. The coefficient of kinetic friction between the trolly and the surface is 0.05.

( $g = 10 m/s^2,$ mass of the string is negligible and no other friction exists ). 

An astronaut accidentally gets separated out of his small spaceship accelerating in interstellar space at a constant rate of 100 $\mathrm{m}/{\mathrm{s}}^2$. What is the acceleration of the astronaut the instant after he is outside the spaceship?

(Assume that there are no nearby stars to exert gravitational force on him.)

A bullet of mass 0.04 kg moving with a speed of 90 m/s enters a heavy fixed wooden block and is stopped after a distance of 60 cm. The average resistive force exerted by the block on the bullet is:

The motion of a particle of mass m is described by y = $\mathrm{ut} + \frac{1}{2}{\mathrm{gt}}^2$. Force acting on the particle is: 

A batsman hits back a ball straight in the direction of the bowler without changing its initial speed of 12 m/s. If the mass of the ball is 0.15 kg, impulse imparted to the ball is:

(Assume linear motion of the ball)

Two identical billiard balls strike a rigid wall with the same speed but at different angles, and get reflected without any change in speed, as shown in fig. The ratio of the magnitudes of impulses imparted to the balls by the wall is:

See the figure given below. A mass of 6 kg is suspended by a rope of length 2 m from the ceiling. A force of 50 N is applied at the mid-point P of the rope in the horizontal direction, as shown. What angle does the rope make with the vertical in equilibrium? (Take g = 10 ${\mathrm{ms}}^{-2}$). Neglect the mass of the rope.

Determine the maximum acceleration of the train in which a box lying on its floor will remain stationary, given that the coefficient of static friction between the box and the train’s floor is 0.15.

See the figure given below, a mass of 4 kg rests on a horizontal plane. The plane is gradually inclined until at an angle θ = 15° with the horizontal, the mass just begins to slide. What is the coefficient of static friction between the block and the surface?

What is the acceleration of the block and tension in the string of the block and trolley system shown in a figure, if the coefficient of kinetic friction between the trolley and the surface is 0.04? (Take g = 10 $\mathrm{m}/{\mathrm{s}}^2$). Neglect the mass of the string.

A cyclist speeding at 18 km/h on a level road takes a sharp circular turn of radius 3 m without reducing the speed. The co-efficient of static friction between the tyres and the road is 0.1. Will the cyclist slip while taking the turn?

A circular racetrack of radius 300 m is banked at an angle of 15°. If the coefficient of friction between the wheels of a race-car and the road is 0.2, what is the (a) optimum speed of the race-car to avoid wear and tear on its tyres, and (b) maximum permissible speed to avoid slipping?

In the figure given below, a wooden block of mass 2 kg rests on a soft horizontal floor. When an iron cylinder of mass 25 kg is placed on top of the block, the floor yields steadily and the block and the cylinder together go down with an acceleration of 0.1 $\mathrm{m}/{\mathrm{s}}^2$. What is the force of the block on the floor after the floor yields? Take g = 10 $\mathrm{m}/{\mathrm{s}}^2$. 

Two identical billiard balls strike a rigid wall with the same speed but at different angles and get reflected without any change in speed as shown in the figure. The direction of the force on the wall due to ball in case (a) and (b) respectively is in the direction of?

The figure shows the position-time graph of a body of mass 0.04 kg. What is the time between two consecutive impulses received by the body? 

Figure shows the position-time graph of a body of mass 0.04 kg. What is the magnitude of each impulse recieved by the  body?

A ball of mass 0.15 kg is dropped from a height 10 m, strikes the ground and rebounds to the same height. The magnitude of impulse imparted to the ball is $\left(\mathrm{g}=10 \mathrm{m}/{\mathrm{s}}^2\right)$ nearly:

Consider the situation shown in the figure. Both the pulleys and the string are light and all the surfaces are frictionless, Find the acceleration of the mass M, and the tension in the string.