Laws of Motion

### A rectangular box lies on a rough inclined surface. The coefficient of friction between the surface and the box is μ. Let the mass of the box be m. a) at what angle of inclination θ of the plane to the horizontal will the box just start to slide down the plane? b) what is the force acting on the box down the plane, if the angle of inclination of the plane is increased to a > θ c) what is the force needed to be applied upwards along the plane to make the box either remain stationary or just move up with uniform speed? d) what is the force needed to be applied upwards along the plane to make the box move up the plane with acceleration a.

a) As the box starts to slide down the plane, $\mu=\tan \theta$ $\theta=\tan ^{-1}(\mu)$ b) If $a>\theta$, the angle of inclination will be the angle of repose and the net force acting will be...

### There are four forces acting at a point P produced by strings as shown in the figure, which is at rest. Find the forces F1 and F2.

Because the point is at rest with a = 0, the resulting forces on the point are zero. As a result, the X and Y axis net components will be zero. It's difficult to resolve all of the forces along the...

### A cricket bowler releases the ball in two different ways a) giving it only horizontal velocity and b) giving it horizontal velocity and a small downward velocity. The speed vs at the time of release is the same. Both are released at a height H from the ground. Which one will have greater speed when the ball hits the ground? Neglect air resistance.

a) $\frac{1}{2} v_{z}^{2}=g H \Rightarrow v_{z}=\sqrt{2 g H}$ Speed at ground is given as: $\sqrt{v_{s}^{2}+v_{z}^{2}}=\sqrt{v_{s}^{2}+2 g H}$ b)$\frac{1}{2} m v_{s}^{2}+m g H$ is the total energy...

### The displacement vector of a particle of mass m is given bya) show that the trajectory is an ellipse b) show that

a) $\begin{array}{l} r(t)=\hat{i} A \operatorname{coswt}+\hat{j} B \sin \omega t \\ x=A \cos \omega t \\ y=B \sin \omega t \\ x / A=\cos \omega t \\ y / B=\sin \omega t \end{array}$ Squaring both...

### There are three forces F1, F2, and F3 acting on a body, all acting on a point P on the body. The body is found to move with uniform speed. a) show that the forces are coplanar b) show that the torque acting on the body about any point due to these three forces is zero

a) The body's acceleration is zero because the resultant force of the three forces F1, F2, and F3 on a location on the body is zero. The directions of forces F1 and F2 are in the plane of the paper,...

### A person in an elevator accelerating upwards with an acceleration of 2 m/s2, tosses a coin vertically upwards with a speed of 20 m/s. After how much time will the coin fall back into his hand?

a = 2 m/s2 is the rising acceleration of an elevator. Gravitational acceleration, g = 10 m/s2 As a result, the net effective acceleration, a' = (a + g) = 12 m/s2, is calculated. In light of the...

### Block A of weight 100N rests on a frictionless inclined plane of slope angle 30o. A flexible cord attached to A passes over a frictionless pulley and is connected to block B of weight W. Find the weight W for which the system is in equilibrium.

Equilibrium between $A$ or $B$, Then we know that, $\mathrm{mg} \sin 30^{\circ}=\mathrm{F}$ $\begin{array}{l} 1 / 2 \mathrm{mg}=\mathrm{F} \\ \mathrm{F}=(1 / 2)(100)=50 \end{array}$ Therefore,...

### In the above-given problem if the lower thread is pulled with a jerk, what happens?

The thread CD breaks when the lower thread is jerked. The thread is inertia at rest, and the mass 2 kg above acts on it.

### A mass of 2 kg is suspended with thread AB. Thread CD of the same type is attached to the other end of 2 kg mass. Lower thread is pulled gradually harder and harder in the downward direction so as to apply force on AB. Which of the threads will break and why?

As the mass 2 kg acts downward, the force acting on the thread AB is equal to the force F. As a result, the force exerted on the AB is 2 kg more than on the D, and the thread AB breaks.

### Why are mountain roads generally made winding upwards rather than going straight up?

Because the force of friction is strong, which decreases the risks of sliding, mountain roads are typically built curving uphill rather than straight up. The value of frictional force increases when...

### A woman throws an object of mass 500 g with a speed of 25 m/s. a) what is the impulse imparted to the object? b) if the object hits a wall and rebounds with half the original speed, what is the change in momentum of the object?

a) Mass of the object, $\mathrm{m}=500 \mathrm{~g}=0.5 \mathrm{~kg}$ $u=0$ $\mathrm{v}=25 \mathrm{~m} / \mathrm{s}$ Impulse is given as: \$ \underset{F}{\vec{F}} \cdot d t=\frac{d \rightarrow}{d...

### Why are porcelain objects wrapped in paper or straw before packing for transportation?

Because porcelain artefacts are delicate, they are wrapped in paper or straw before being packed for shipment. When the acceleration (v – u)/t drops while carrying these items, the shift in velocity...

### A body of mass 10 kg is acted upon by two perpendicular forces, 6N and 8N. The resultant acceleration of the body is a) 1 m/s2 at an angle of tan-1 (4/3) w.r.t 6N force b) 0.2 m/s2 at an angle of tan-1 (4/3) w.r.t 6N force c) 1 m/s2 at an angle of tan-1(3/4) w.r.t 8N force d) 0.2 m/s2 at an angle of tan-1(3/4) w.r.t 8N force

The correct answers are a) 1 m/s2 at an angle of tan-1 (4/3) w.r.t 6N force c) 1 m/s2 at an angle of tan-1(3/4) w.r.t 8N force

### A hockey player is moving northward and suddenly turns westward with the same speed to avoid an opponent. The force that acts on the player is a) frictional force along westward b) muscle force along southward c) frictional force along south-west d) muscle force along south-west

The correct answer is c) frictional force along the south-west

### One end of a string of length l is connected to a particle of mass m and the other to a small peg on a smooth horizontal table. If the particle moves in a circle with speed v the net force on the particle (directed towards the centre) is : (i) T (ii) T – mv2/l (iii) T + mv2/l (iv) 0 T is the tension in the string. [Choose the correct alternative].

T is the particle's net force, and it is directed towards the centre. It gives the particle the centrifugal force it needs to travel in a circle.

### A train runs along an unbanked circular track of radius of 30 m at a speed of 54 km/h. The mass of the train is 106 kg. What provides the centripetal force required for this purpose the engine or the rails? What is the angle of banking required to prevent wearing out of the rail?

The track's radius is 30 metres. The train's speed = 54 km/h = 54 x (5/18) = 15 m/s The train's mass is 106 kg. The force of lateral friction created by the rails on the train wheels provides the...

### Figure shows a man standing stationary with respect to a horizontal conveyor belt that is accelerating with 1 ms-2. What is the net force on the man? If the coefficient of static friction, between the man’s shoes and the belt is 0.2, up to what acceleration of the belt can the man continue to be stationary relative to the belt? (Mass of the man = 65 kg.)

Here, the conveyor belt's acceleration is a = 1 ms-2. s=0.2 is the static friction coefficient. m = 65 kg m = 65 kg m = 65 kg m = 65 kg m = 65 kg Ma = 65 x 1 = 65N is the Net Force. The friction...

### Figure shows the position-time graph of a particle of mass 0.04 kg. Suggest a suitable physical context for this motion. What is the time between two consecutive impulses received by the particle? What is the magnitude of each impulse?

This graph might represent a ball rebounding between two walls separated by two centimetres. Every two seconds, the ball bounces back and forth between the walls at a constant pace. 2 x 10-2)/2 =...

### Explain why a cricketer moves his hands backwards while holding a catch.

The ball will have a lot of momentum when the batsman smashes it. When he slides his hands rearward, the contact duration increases, and the force decreases.

### Explain why passengers are thrown forward from their seats when a speeding bus stops suddenly,

Because of the inertia of motion. When a bus abruptly stops, the bottom half of a person's body in touch with the seat comes to a halt, while the upper part of the body continues to move. As a...

### Explain why a horse cannot pull a cart and run in empty space,

When the horse pulls the cart, it exerts a certain amount of force on the ground. The ground will exert an equal and opposite reaction force on the horse's feet if the third rule of motion is...

### A stone of mass 0.25 kg tied to the end of a string is whirled round in a circle of radius 1.5 m with a speed of 40 rev./min in a horizontal plane. What is the tension in the string? What is the maximum speed with which the stone can be whirled around if the string can withstand a maximum tension of 200 N?

The stone weighs 0.25 kilogramme. r = 1.5 m Radius n= 40/60 = (23) rev/sec is the number of revolutions per second. = 2n = 2 x 3.14 x (23) is the angular velocity. The centripetal force is provided...

### A batsman deflects a ball by an angle of 45° without changing its initial speed which is equal to 54 km/h. What is the impulse imparted to the ball? (Mass of the ball is 0.15 kg.)

The ball's speed is 54 km/h. The ball is deflected backwards to a total angle of 450 degrees. The initial momentum of the ball is mucosӨ = (0.15 x 54 x 1000 x cos 22. 5)/3600 = 0.15 x 15 x 0.9239...

### Two bodies of masses 10 kg and 20 kg respectively kept on a smooth, horizontal surface are tied to the ends of a tight string. A horizontal force F = 600 N is applied to (i) A, (ii) B along the direction of string. What is the tension in the string in each case?

Given, A body mass of 10 kilogrames (m1) B, m2 = 20 kg, 600 N horizontal force m = m1 + m2 = 30 kg is the total mass of the system. Using Newton's second rule of motion, we can calculate ma = f...

### Figure shows the position-time graph of a particle of mass 4 kg. What is the (a) force on the particle for t 4 s, 0 < t < 4 s? (b) impulse at t = 0 and t = 4 s? (Consider one-dimensional motion only).

When t<0, the particle's distance travelled is zero. As a result, the particle's force is zero. When 0< t< 4s occurs, the particle is travelling at a constant speed. As a result, there will...

### . A man of mass 70 kg, stands on a weighing machine in a lift, which is moving (a) upwards with a uniform speed of 10 ms-1. (b) downwards with a uniform acceleration of 5 ms-2. (c) upwards with a uniform acceleration of 5 ms-2. What would be the readings on the scale in each case? (d) What would be the reading if the lift mechanism failed and it hurtled down freely under gravity?

m = 70 kg 10 m/s2 = g In each scenario, the weighing machine measures the response R, or apparent weight. (a) The lift's acceleration equals 0 when it travels upwards at a uniform speed of 10 m/s. R...

### A bob of mass 0.1 kg hung from the ceiling of a room by a string 2 m long is set into oscillation. The speed of the bob at its mean position is 1 ms-1. What is the trajectory of the bob if the string is cut when the bob is at one of its extreme positions?

(a) The velocity of the bob is zero when it is at one of its extreme positions. If the string is severed, the bob will fall vertically downward due to its weight F = mg.

### A truck starts from rest and accelerates uniformly at 2.0 ms-2. At t = 10 s, a stone is dropped by a person standing on the top of the truck (6 m high from the ground). What are the (a) velocity, and (b) acceleration of the stone at t = 11s? (Neglect air resistance.)

u = 0 is the initial velocity. a = 2 ms-2, a = 2 ms-2, a = 2 ms-2, a = 2 m   t=10s t=10s t=10s t=10   We get v = u + at using the equation v = u + at.   20 m/s = v = 0 + 2 x 10  ...

### A body of mass 0.40 kg moving initially with a constant speed of 10 ms-1 to the north is subject to a constant force of 8.0 N directed towards the south for 30 s. Take the instant the force is applied to be t = 0, the position of the body at that time to be x = 0, and predict its position at t = -5 s, 25 s, 100 s.

Given, Body mass is 0.40 kg. u = 10 m/s initial velocity f = -8 N force (retarding force) Using the formula S = ut + (12) at2, (a) At time t = – 5 s, position From t = 0 s, the force acts on the...

### The driver of a three-wheeler moving at a speed of 36 km/h sees a child standing in the middle of the road and brings his vehicle to rest in 4.0 s just in time to save the child. What is the average retarding force on the vehicle? The mass of the three-wheeler is 400 kg, and the mass of the driver is 65 kg.

Given, u=36 km/h is the initial velocity. v = 0 is the final velocity. The three-mass wheeler's is m1=400 kg. The driver's mass is m2 = 65 kg. The time it took to bring the car to a complete stop...

### . A constant retarding force of 50 N is applied to a body of mass 20 kg moving initially with a speed of 15 m s-1. How long does the body take to stop?

Here, – 50 N force (since it is a retarding force) m = 20 kg mass 0 = v u = 15 m s-1 u = 15 m s-1 u = 15 m F = ma force a = F/m = -50/20 = – 2.5 ms-2 v = u + at is the equation to use. 0 = 15 + (-...

### Give the magnitude and direction of the net force acting on a stone of mass 0.1 kg, (a) just after it is dropped from the window of a stationary train (b) just after it is dropped from the window of a train running at a constant velocity of 36 km/h (c ) just after it is dropped from the window of a train accelerating with1 m s-2 (d) lying on the floor of a train which is accelerating with 1 m s-2, the stone being at rest relative to the train. Neglect air resistance throughout.

(a) Stone mass = 0.1 kg 10 ms^(-2)= acceleration   F = mg = 0.1 x 10 = 1.0 N is the net force.   The force is applied vertically and downwards.   (b) The train maintains a steady...