Tiêu đề 3. Laws of Motion.pdf ✅

Laws of Motion 41 3 Laws of Motion QUICK LOOK Equilibrium of Concurrent Force If all the forces working on a body are acting on the same point, then they are said to be concurrent. A body, under the action of concurrent forces, is said to be in equilibrium, when there is no change in the state of rest or of uniform motion along a straight line. The necessary condition for the equilibrium of a body under the action of concurrent forces is that the vector sum of all the forces acting on the body must be zero. Mathematically for equilibrium 0 ∑Fnet = or 0; ∑Fx = 0; 0. ∑ ∑ F F y z = = Three concurrent forces will be in equilibrium, if they can be represented completely by three sides of a triangle taken in order. Figure: 3.1 Three Concurrent Forces Lami’s Theorem: For concurrent forces 1 2 3 sin sin sin F F F α β γ = = Figure: 3.2 Three Concurrent Forces in Equilibrium If a point is under action of several forces, then for equilibrium of point, net force 1 2 Σ = + + = F F F ... 0 Equilibrium: If a point is under action of several forces, then for equilibrium of point, net force 1 2 Σ = + + = F F F 0 ... Stable equilibrium: If on slight displacement from equilibrium position a body has tendency to regain its original position, it is said to be in stable equilibrium. In case of stable equilibrium potential energy is minimum 2 2 d U ve dr     = +   and so centre of gravity is lowest. Fig, Figure: 3.3 Unstable Equilibrium: If on slight displacement form equilibrium position body moves in the direction of displacement, the equilibrium is said to be unstable. In this situation potential energy of the body is maximum 2 2 d U ve dr     = −   and so centre of gravity is highest. Figure: 3.4 Neutral equilibrium: If on slight displacement from equilibrium position a body has no tendency to come back to original position or to move in the direction of displacement, it is said to be in neutral equilibrium. In this situation potential energy of the body is constant 2 2 0 d U dr     =   and so centre of gravity remains at constant height. Figure: 3.5 Newton's Laws of Motion Newton’s First Law: A body continues to be in its state of rest or of uniform motion along a straight line, unless it is acted upon by some external force to change the state. If no net force acts on a body, then the velocity of the body cannot change i.e. the body cannot accelerate. Newton’s first law defines inertia and is rightly called the law of inertia. Body (1) (2) (3) (2) M D M (1) D (3) M M D D M (1) (3) (2) D M D M M D D α β γ F1 F2 F3 A B C F1 F2 F3
42 Quick Revision NCERT-PHYSICS Newton’s I law, if external F v = = 0, 0 or constant Inertia is of three types: Inertia of rest, Inertia of motion, Inertia of direction Inertia of rest: It is the inability of a body to change by itself, its state of rest. This means a body at rest remains at rest and cannot start moving by its own. Inertia of Motion: It is the inability of a body to change itself its state of uniform motion i.e., a body in uniform motion can neither accelerate nor retard by its own. Inertia of Direction: It is the inability of a body to change by itself direction of motion. Note If the motion of the bus is slow, the inertia of motion will be transmitted to the body of the person uniformly and so the entire body of the person will come in motion with the bus and the person will not experience any jerk. When a horse starts suddenly, the rider tends to fall backward on account of inertia of rest of upper part of the body as explained above. If we place a coin on smooth piece of card board covering a glass and strike the card board piece suddenly with a finger. The cardboard slips away and the coin falls into the glass due to inertia of rest. The dust particles in a durree fall off when it is beaten with a stick. This is because the beating sets the durree in motion whereas the dust particles tend to remain at rest and hence separate. Newton’s Second Law: The rate of change of linear momentum of a body is directly proportional to the external force applied on the body and this change takes place always in the direction of the applied force. If a body of mass m, moves with velocity v then its linear momentum can be given by p mv = and if force F is applied on a body, then dp dp F F K dt dt ∝ ⇒ = or dp F dt = (K = 1 in C.G.S. and S.I. units) or ( ) d dv F mv m ma dt dt = = = (As == dt vd a acceleration produced in the body) ∴ F ma = Force = mass × acceleration Newton’s II law d p F ma dt = = Where p mu = Linear momentum Newton’s Third Law: To every action, there is always an equal (in magnitude) and opposite (in direction) reaction. When a body exerts a force on any other body, the second body also exerts an equal and opposite force on the first. Forces in nature always occurs in pairs. A single isolated force is not possible. Any agent, applying a force also experiences a force of equal magnitude but in opposite direction. The force applied by the agent is called ‘Action’ and the counter force experienced by it is called ‘Reaction’. Action and reaction never act on the same body. If it were so the total force on a body would have always been zero i.e. the body will always remain in equilibrium. If FAB = force exerted on body A by body B (Action) and FBA = force exerted on body B by body A (Reaction) Then according to Newton’s third law of motion F F AB BA = − Newton’s III law F F 12 21 = − Modification of Newton’s Laws of Motion: According to Newton, direction and time i.e., time and space are absolute. The velocity of observer has no effect on it. But, according to special theory of relativity Newton’s laws are true, as long as we are dealing with velocities which are small compare to velocity of light. Hence the time and space measured by two observers in relative motion are not same. Some conclusions drawn by the special theory of relativity about mass, time and distance which are as follows: Let the length of a rod at rest with respect to an observer is 0 L .If the rod moves with velocity v w.r.t. observer and its length is L, then 2 2 0 L L v c = −1 / where, c is the velocity of light. Now, as v increases L decreases, hence the length will appear shrinking. Let a clock reads T0 for an observer at rest. If the clock moves with velocity v and clock reads T with respect to observer, then 0 2 2 1 T T v c = − Hence, the clock in motion will appear slow. Let the mass of a body is m0 at rest with respect to an observer. Now, the body moves with velocity v with respect
Laws of Motion 43 to observer and its mass is m, then 0 0 2 2 1 m m m v c = − is called the rest mass. Hence, the mass increases with the increases of velocity. Note If v << c, i.e., velocity of the body is very small w.r.t. velocity of light, then m m= 0 i.e., in the practice there will be no change in the mass. If v is comparable to c, then 0 m m> , i.e., mass will increase. If v c = , then 0 2 2 1 m m v v = − or 0 0 m m = = ∞ Hence, the mass becomes infinite, which is not possible, thus the speed cannot be equal to the velocity of light. The velocity of particles can be accelerated up to a certain limit. In cyclotron the speed of charged particles cannot be increased beyond a certain limit. Apparent Weight of a Body In a Lift When the lift is at rest or moving with uniform velocity i.e., a = 0, mg R R mg − = = 0, orW W app = 0 Where,W R app = = reaction of supporting surface and W mg 0 = = true weight. Figure: 3.6 When the lift moves upwards with an acceleration a: R mg ma = = ( ) 1 a R m g a mg g   = + = +     ∴ 0 1 app a W W g   = +     When the lift moves downwards with an acceleration a: mg R ma − = or ( ) 1 a R m g a mg g   = − = −     ∴ 0 1 app a W W g   = −     Here, if , app a g W > will be negative. Negative apparent weight will mean that the body is pressed against the roof of the lift instead of floor. When the lift falls freely, i.e., a=g: R mg g g = − = ( ) 0 or 0 Wapp = . If the carriage/lift begins to fall freely, then the tension in the string becomes zero. Mass m experiences a pseudo force ma opposite to acceleration; the mass m is in equilibrium inside the carriage and T ma sin , θ = T mg cos , θ = i.e., 2 2 T m g a = + the string does not remain vertical but inclines to the vertical at angle 1 θ tan ( / ) a g − = opposite to acceleration. This arrangement is called accelerometer and can be used to determine the acceleration of a moving carriage from inside by noting the deviation of a plumb line suspended from it form the vertical. Impulse When a large force works on a body for very small time interval, it is called impulsive force. An impulsive force does not remain constant, but changes first from zero to maximum and then from maximum to zero. In such case we measure the total effect of force. Impulse of a force is a measure of total effect of force. ∫ = 2 1 t t dtFI . Impulse is a vector quantity and its direction is same as that of force. Dimension: [MLT–1] Units: Newton-second or Kg-m- −1 s (S.I.) and Dyne-second or gm-cm- −1 s (C.G.S.) Force-time graph: Impulse is equal to the area under F-t curve. Figure: 3.7 Force vs. Time Force Time t F Freeling falling acc. a = 9.8 m/s2 Coming down with acc.a u – constant a = 0 moving upward with acc.a
44 Quick Revision NCERT-PHYSICS If we plot a graph between force and time, the area under the curve and time axis gives the value of impulse. I = Area between curve and time axis 1 2 = × Base × Height 1 2 = F t If Fav is the average magnitude of the force then tFdtFdtFI av t t av t t == ∆= ∫∫ 2 1 2 1 Figure: 3.8 F-t Curve From Newton’s second law dp F dt = 2 2 1 1 t p t p F dt dp = ∫ ∫ ⇒ 2 1 I p p p = − = ∆ i.e. The impulse of a force is equal to the change in momentum. This statement is known as Impulse momentum theorem Monkey Climbing a Rope: Let T be the tension in the rope. Figure 3.9 When the monkey climbs up with uniform speed: T mg = When the monkey moves up with an acceleration a:T mg ma − = or T mg g a = + ( ) When the monkey moves down with an acceleration a: mg T ma − = or T mg g a = − ( ) Friction Figure 3.10 Force of friction arises due to molecular interaction of two bodies. Limiting static friction is maximum and rolling friction is minimum. Figure: 3.11 Frictional force F N = μ where nor R = normal reaction and acts opposite to motion. Static frictional force is self adjusting. When a body is at rest, then frictional force = applied force; but limiting frictional force μsN. If F is limiting friction and N or R is normal reaction, then angle of friction λis given by tan s F R λ μ = = ⇒ ( ) 1 tan λ μs − = . Angle of repose θis the angle of inclined plane at which the body is just at the average of sliding and is given by ( ) 1 tan θ λ μs − = = Graph between Applied Force and Force of Friction Figure: 3.12 Applied Force vs. Force of Friction Part OA of the curve represents static friction ( ). Fs Its value increases linearly with the applied force At point A the static friction is maximum. This represent limiting friction ( ). Fl Beyond A, the force of friction is seen to decrease slightly. The portion BC of the curve therefore represents the kinetic friction ( ). Fk As the portion BC of the curve is parallel to x-axis therefore kinetic friction does not change with the applied force, it remains constant, whatever be the applied force. A B C Fl Fk Force of friction Applied force O Fs y x W N N f W f d F Fraction T Mg t F Fav t1 t2 ∆t Impulse