Laws of Motion Notes PDF 2024-2025

M.E.S INDIAN SCHOOL, DOHA - QATAR Notes 2024 - 2025 Section : Boys/Girls Date :17-05-2024 Class & Div. : XI (All Divisions) Subject : Physics L...

M.E.S INDIAN SCHOOL, DOHA - QATAR Notes 2024 - 2025 Section : Boys/Girls Date :17-05-2024 Class & Div. : XI (All Divisions) Subject : Physics Lesson / Topic: LAWS OF MOTION xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Force: It is an external push or pull which either changes or tries to change the position of rest or of uniform motion of a body. Inertia: It is the property of a body by virtue of which it opposes the change in position of rest or of uniform motion in absence of external force. Inertia of a body is directly proportional to its mass. Newton’s First Law of Motion: A body at rest remains at rest and a body is in motion continues its motion in a straight line until it is compelled by an external unbalanced force. Momentum: Momentum of a body is defined as the product of its mass and velocity. ⃗ = mv p ⃗ −1 Its SI unit is Kg m s It is a vector quantity. The direction of momentum is in the direction of velocity. Newton’s Second Law of Motion: It states that the rate of change of momentum of an object is proportional to the applied unbalanced force in the direction of force. Consider a body of mass m moving with a velocity u, when a force F is applied along the direction of motion, its velocity increases to v. According to Newton’s Second Law of Motion; Change in Momentum Force ∝ Time mv − mu F=k t k=1 mv − mu ∴F= t v−u F = m( ) t v−u 𝐅 = 𝐦𝐚 (∵ = a) t SI unit of force is newton (N) 1N = 1 kg x 1 ms−2 1 F 061, Rev 01, dtd 10th March 2020 Kg wt is the unit of gravitational force 1 kg wt is the force with which earth attracts a body of mass 1 kg towards its centre. 1 kg wt = 9.8 N Newton’s Third Law of Motion: To every action there is an equal and opposite reaction. Action and reaction forces are always equal in magnitude, opposite in direction and act on different bodies. Therefore they do not cancel each other. Impulsive Force: The force acting for a very short time is called impulsive force. It is measured as the rate of change of momentum. Example: Force exerted on a ball when it is hit by a bat. Force exerted on a bullet when it is fired from a gun. The effect of impulsive force is called impulse. Impulse (I): Impulse of a force is the product of the force and the time during which the force acts on a body. Impulse = Force x Time I =Fxt Its SI unit is NS or kg ms−1 To show that, Impulse = Change on momentum According to Newton’s Second Law, mv − mu F= t F. t = mv − mu I = Change in Momentum (∵ I = F x t) Isolated System: A system is said to be isolated if there is no exchange of mass and energy between the system and surroundings. Law of conservation of linear momentum: It states that in the absence of external force the total linear momentum of an isolated system remains same or conserved. To prove the law of conservation of momentum by using Newton’s Second Law Consider an isolated system consists of n particles of masses m1 , m2 , m3 , … … … … mn moving with velocities v1 , v2 , v3 , … … … … vn The total linear momentum is given by p = m1 v1 + m2 v2 + m3 v3 … … … … mn vn If an external force F acts on the system According to Newton’s Second Law, 2 F 061, Rev 01, dtd 10th March 2020  dp Fext = dt When Fext = 0 dp =0 dt p = constant Therefore, the total momentum of the system remains constant in the absence of external force. To prove the law of conservation of momentum by using Newton’s Third Law Consider two bodies A and B of masses m1 and m2 moving with velocities u1 and u2 along a straight line in the same direction. After collision, let them move with velocities v1 and v2 in the same direction. Let t be the time of collision. Force exerted by A on B m2 v2 − m2 u2 FBA = (Action) t Force exerted by B on A m1 v1 − m1 u1 FAB = (Reaction) t According to Newton’s Third Law Action = - Reaction FBA = −FAB m2 v2 − m2 u2 = −m1 v1 + m1 u1 m1 v1 + m2 v2 = m1 u1 + m2 u2 Total Momentum after collision = Total Momentum before collision i.e., total momentum remains constant when Fext = 0. Recoil of a gun: When a shot is fired from a gun, the shot acquires a large forward momentum due to explosion of the gun powder. Consequently the gun gets an equal backward momentum and the gun is pushed backward with a velocity V. This backward motion of the gun is called recoil of the gun. The velocity V of the gun just after the firing is called the recoil velocity of the gun; the velocity v of the bullet just after the firing is called the muzzle velocity of the bullet. Let M be the mass of the gun and m that of the shot. Before firing both are at rest. After firing let v be the velocity of the shot and V that of the gun. By law of conservation of linear momentum. Total Momentum after firing = Total Momentum before firing mv + MV = 0 MV = −mv 3 th F 061, Rev 01, dtd 10 March 2020  −𝐦𝐯 𝐕= 𝐌 This is called recoil velocity of gun. Concurrent forces: Forces acting at the same point on a body are called concurrent forces. Forces ⃗⃗⃗⃗ F1 , ⃗⃗⃗⃗ F2 , ⃗⃗⃗⃗ F3 , ⃗⃗⃗⃗ F4 and ⃗⃗⃗⃗ F5 acting at a point O are called concurrent forces. The resultant of these forces is equal to their vector sum ⃗F = ⃗⃗⃗⃗ F1 + ⃗⃗⃗⃗F2 + ⃗⃗⃗⃗ F3 + ⃗⃗⃗⃗ F4 + ⃗⃗⃗⃗ F5 If the magnitude of the resultant is equal to zero, the point O is the equilibrium. Coplanar Concurrent Forces: When forces P, Q and R acting at a point O are in the same plane, these forces are called coplanar concurrent forces. Equilibrium of Concurrent Forces: The point O will be in equilibrium if one of the following laws is satisfied - (i) The vector sum of ⃗P and ⃗Q ⃗ must be equal and opposite to ⃗R. ⃗ +Q P ⃗⃗ = −R⃗ (ii) If P, Q and R can be represented by the sides of the triangle taken in order, O will be in equilibrium. P Q R = = = constant LN ML NM (iii) Each force must be proportional to the sine of the angle between the other two ⃗P ⃗Q ⃗ ⃗R = = = constant sin α sin β sin γ This is called Lami’s theorem. General condition of equilibrium of a point acted upon by a number of coplanar forces (i) The sum of the resolved components of all forces in the X-direction must be zero. ⃗𝑥=0 i.e., ∑ F 4 F 061, Rev 01, dtd 10th March 2020  (ii) The sum of the resolved components of all forces in the Y-direction must be zero. ⃗𝑦 =0 i.e., ∑ F Friction: Friction is the force which opposes relative motion between two surfaces in contact with each other. Types of friction: (i) Static friction – The force of friction between the two surfaces in contact before one body actually starts moving on the surface of another. (ii) Limiting friction (𝐟𝐬 ) – The force of friction between two bodies in contact when one body is in position to just begins to slide over the surface of another body is called limiting friction or static limiting friction. (iii) Kinetic friction (𝐟𝐤 ) – The force of friction between the surfaces in contact when one body is in motion on the surface of another is called kinetic friction or dynamic friction. Laws of limiting friction: (i) The magnitude of the limiting static friction is independent of the area, when the normal reaction between the surfaces remains same. (ii) The limiting static friction (fs ) is directly proportional to normal reaction (R). fs ∝ R fs = µs R µs is constant called coefficient of static limiting friction. 𝐟𝐬 µ𝐬 = 𝐑 Laws of Kinetic friction: (i) Kinetic friction has a constant value depending on the nature of two surfaces in contact and independent of the area. (ii) Kinetic friction (fk ) is proportional to normal reaction (R) fk ∝ R fk = µk R µk is a constant called coefficient of kinetic friction 𝐟𝐤 µ𝐤 = 𝐑 (iii) The Kinetic Friction between two surfaces is independent of the relative velocity between the surfaces. Rolling friction: Rolling friction between two surfaces in contact is the force of friction which comes into play when a body rolls over another. 5 F 061, Rev 01, dtd 10th March 2020 Friction is necessary evil Advantages of Friction: (i) Without friction between our feet and the ground, it will not be possible to walk. (ii) The tyres of the vehicles are made rough to increase friction. (iii) Various parts of machine are able to rotate due to friction between belt and pulley. (iv) It would be impossible to climb, to fix a nail if there were no friction. Disadvantages of Friction: (i) Wear and tear of the machinery is due to friction. (ii) Friction between different parts of the rotating machines produces heat and causes damage to them. (iii) We have to apply extra power to machines in order to overcome friction. Thus, the efficiency of machines decreases. Methods of Reducing Friction: (i) Polishing (ii) Lubricating (iii) Stream Lining (iv) Avoiding Moisture (v) Use of Ball Bearings Lubrication A lubricant is a substance which forms a thin layer between two surfaces in contact. It fills the depressions present on the surfaces of contact and hence friction is reduced. Generally oil and grease are used as lubricants to reduce friction as well as to protect the moving parts from overheating. A modern lubricant is a mixture of mineral oil, vegetable oil and colloidal thin oil. For heavy machinery thick oil or grease is required because thin oil would be squeezed out from between the surfaces in contact by great pressure. In very heavy machinery solid lubricant graphite is used. For light machinery like watches, sewing machines etc, thin oil is used. Flow of compressed pure dry air acts as lubricant. It reduces friction between the moving parts by providing an elastic cushion. It also prevents overheating and settling of dust on moving parts. Centripetal force: When a particle moves round a circle there is a centripetal acceleration. If v is the speed of the particle and r the radius of the path, v2 Centripetal acceleration = = rω2 r The force acting on the body that gives this acceleration is called centripetal force. Centripetal Force = mass x centripetal acceleration mv 2 F= = mrω2 r It is this force that keeps the body in a circular path by continuously changing its direction. On removing this force the body will fly off along the tangent with constant velocity. When a body is moving round the circle with a uniform speed, no work is done as there is no displacement in the direction of the force. Examples of Centripetal Force 1) When a body tied to one end of a string is whirled in a horizontal circle, the necessary centripetal force is supplied by the tension T in the string. 2) For the earth moving round the sun, centripetal force is provided by the gravitational force of 6 attraction between the earth and the sun. F 061, Rev 01, dtd 10th March 2020 Centrifugal reaction The centripetal force produces a centrifugal reaction. When we whirl a stone tied to the end of a string, the centripetal force necessary to keep the stone in the circular path is provided by the tension in the string. The hand in turn will experience an equal reaction force outward along the radius. It must be noted that centripetal force acts on the rotating body while the centrifugal reaction force acts on the body at the centre. Since these two forces act on different bodies they do not cancel each other. Examples in circular motion Car on a circular level road: Let a car of mass M move with constant speed v on a flat horizontal circular portion of a road of radius r. The forces acting on the car are 1) The weight Mg of the car acting vertically down, which is balanced by the normal reaction R of the ground acting vertically upwards ∴ R = Mg Mv2 2) Centripetal force( ) acting on the car, directed towards the centre of the circle. This force is r provided by the force of friction between the car tyre and the road. If F is the force of friction and µs is the coefficient of friction F = µs R = µs Mg For safe turning the centripetal force should be equal or less than the available force of friction. Mv 2 Mv 2 ≤ F (or) ≤ µs Mg r r v 2 ≤ µs rg ∴ vmax = √μrg This is the maximum speed limit with which a car can have safe turning. It the speed of the car exceeds this value, the force due to friction will not be sufficient to provide the centripetal force and it will skid off the road. Banking of roads: The maximum speed limit a vehicle can have on a curved level road depends on ‘µ’ the coefficient of friction between the tyre and the road. The force of friction is not a reliable source for providing the required centripetal force for the vehicle. The friction causes wear and tear of the tyre. Moreover, if the speed exceeds the limit it may cause accident by skidding. To avoid this, the road bed is banked at curves. At curves the surface of the road is so laid that it slopes with the outer edge raised above the inner edge. This is called banking of roads. The angle between the surface of the road and the horizontal is called banking angle. 7 F 061, Rev 01, dtd 10th March 2020 Let r be the radius of curvature of the road and θ the angle of banking and Mg the weight of the vehicle. The normal reaction R can be resolved into two components, R cos θ – vertically upwards R sin θ – towards the centre of curvature of the curve R cos θ = Mg and R sin θ gives the necessary centripetal force. Mv 2 R sin θ = r R sin θ Mv 2 = ÷ Mg R cos θ r v2 tan θ = rg The speed limit at which the curve can be negotiated without wear and tear of the tyre is given by v = √rg tan θ If the force of friction is also taken into account it can be shown that the safe limit of speed in this case is rg(tan θ + μs ) vmax = √ 1 − μs tan θ 8 F 061, Rev 01, dtd 10th March 2020