Newton s Laws of Motion and Gravitation

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1 Newton s Laws of Motion and Gravitation Introduction: In Newton s first law we have discussed the equilibrium condition for a particle and seen that when the resultant force acting on the particle is zero, it is in equilibrium and, of course, the acceleration of the body is also zero. The next logical step is to ask how a body behaves when the resultant force on it is not zero. The answer to this question is contained in second law, which shows that when the resultant force is not zero the body moves with accelerated motion, and that with a given force, the acceleration depends on a property of the body known as mass. Newton s first law of motion: According to Newton's first law An object at rest will remain at rest and an object in motion continues in motion with the same speed and in the same direction unless it is acted upon by an unbalanced force. This law is often called the law of inertia". This means that there is a natural tendency of objects to keep on doing what they're doing. All objects resist changes in their state of motion. In the absence of an unbalanced force, an object in motion will maintain this state of motion. The equilibrium condition of a body: If an object is not accelerating, it is in equilibrium. Newton's First Law says that" if the net force on an object is zero, it will be in equilibrium - it won't accelerate". The converse of this statement is also true - "If an object is in equilibrium (is not accelerating), then the net force on it must be zero." Suppose there is a book lying (at rest) on a table. The book is not accelerating - in other words, it is in equilibrium. Since it is in equilibrium, the net force on the book is zero. The Earth is pulling downward on the book with a force we call the book's weight. Suppose the book's weight is 1 N. If this were the only force on the book, there would be a net force on the book, and the book would not be in equilibrium. The table must be exerting an upward force on the book of exactly 1 N in order to balance (or cancel) the weight. This force, exerted by the table on the book, is called a support force or normal force. Newton s second law of motion: We know, from experience, that an object at rest never starts to move by itself; a push or pull must be exerted on it by some other body. Similarly, a force is required to slow down or to stop a body already in motion, and to make a moving body deviate from straight line motion requires a sideway force. All these processes (speeding up, slowing down, or changing direction) involve a change in either the magnitude or direction of the velocity. Thus in each case the body accelerates, and an external force must act on it to produce the acceleration. The second law discusses about these factors and states that The rate of change of momentum of a body is directly proportional to the applied force and acts in the direction of the force. Mathematically this law can be represented as dp F dt where p momentum mv massvelocity of the body PHY 11/Chapter-3 1

2 dp F C (1) dt Where C = constant and is chosen as C =1. Then we can write from (1) dp d dv F ( mv) m ma dt dt dt Therefore F ma. () If we consider only magnitudes, then F = ma Therefore F m a Concept of inertial mass: From this relation the concept of inertial mass of a body can be defined. If we apply a force to a body it will accelerate i. e. acceleration is produced in the body. The ratio of the applied force to the corresponding acceleration is called the mass. This definition of mass is termed as the definition of the inertial mass and, in fact, it is purely a mathematical concept. Newton s third law of motion: Newton s third law states that For every action there is an equal and opposite reaction. This means that for every force there is a reaction force that is equal in size, but opposite in direction. That is to say that whenever an object pushes another object it gets pushed back in the opposite direction equally hard. If a body acts a force F 1 upon another body, the second body will also act a force F upon the first body. According to Newton s third law, F 1 = - F. Here, F 1 is the action and F is the reaction of that action. Newton s law of gravitation: The law of universal gravitation was described by Newton and may be stated as, Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particle and inversely proportional to the square of the distance between them. Suppose m 1, m are the masses of two particles and d is the distance between them, then according the law of gravitation, 1 m 1 m & d Therefore, m1m G (3) d where is the gravitational force on either particle and G is a constant known as universal gravitational constant. Concept of gravitational mass: We have defined mass of a body as a mathematical concept which is called inertial mass. Gravitational mass is the mass of an object measured using the effect of a PHY 11/Chapter-3

3 gravitational field on the object. Gravitational mass is measured by comparing the force of gravity of an unknown mass to the force of gravity of a known mass. If the earth were a homogeneous sphere of mass m E, the force exerted by it on a small body of mass m, at a distance r from the center, would be mm F E g G r, This small body of mass m is called the gravitational mass. The above equation provides that the body lies outside the earth i. e. r is greater than the radius of the earth. Thus it is also shown that the gravitational force exerted on or by a homogeneous sphere is the same as if the entire mass of the sphere were concentrated in a point at its center. At points inside the earth, these statements need to be modified. The force would be found to decrease as the center is approached, rather than increasing as 1/r. This occurred because, as the body enters the interior of the earth, some of the earth s mass is on the side of the body opposite from the center of the earth and pulls the body in the opposite direction. Exactly at the center of the earth, the gravitational force on the body is zero. Weight: The weight of a body can be defined as the resultant gravitational force exerted on the body by all other bodies in the universe. If the earth were a homogeneous sphere of radius and mass m E, the weight W of a small body of mass m at or near its surface would be GmmE W F g (4) When a body is allowed to fall freely, the force accelerating it is its weight W, and the acceleration produced by this force is the acceleration due gravity g. The general relation F=ma therefore becomes, for the special case of a freely falling body W mg. (5) From (4) and (5), it follows that GmE g.. (6) Equation (6) shows that the acceleration due to gravity is the same for all bodies and very nearly constant. Applications of Newton s law: Example 1: A block whose mass is 1 kg rests on a horizontal table. What constant horizontal force T is required to give it a velocity of 4 ms -1 in s. Assume that the block starts from rest, the friction between the surface of table and block is 5N. The acceleration is 1 v v 4ms a ms t s f T The resultant of the force along x-axis is W PHY 11/Chapter-3 3

4 F T f ma x T max f (1kg )(ms ) 5N 5N Therefore the horizontal force is 5N. x Example : An elevator and its load have a total mass of 8kg. Find the tension T in the supporting cable when the elevator, originally moving downward at 1ms -1, is brought to rest with constant acceleration in a distance of 5m. The acceleration can be obtained from v v ay Here v and y is negative since they are measured in downward direction. 1 v v ( 1ms ) a ms y ( 5m) The acceleration is positive and therefore in upward direction. The resultant force is F T W where W is the weight of the elevator. Therefore the tension T is T F W ma mg m( a g) 8kg ms 9.8ms T 944N Therefore the tension is 944N. The tension T must be greater than the weight W to cause the upward acceleration while the elevator is stopping. Exercise 4-1: a) what is the mass of a body that weighs 1N at a point where g = 9.8ms - b) weighs 1dyne at a point where g = 98cms -? a) We know, W mg W 1N m.1kg. g 9.8ms W 1dyne b) m.1g. g 98cms Exercise 4-5: A constant horizontal force of 4N acts on a body on a smooth horizontal table. The body starts from the rest and is observed to move 1m in 5s. a) What is the mass of the body? b) If the force ceases to act at the end of 5s, how far will the body move in the next 5s? a) The acceleration of the body can be obtained from 1 x x v t at T 8kg W PHY 11/Chapter-3 4

5 x, v, t 5s, x 1m,4N Here, x (1m) a 8ms t (5s) Therefore, the mass of the body is F ma F N m 4 5kg a 8ms b) If the force is ceased the body will move in constant speed. Just before the ceasing of force the velocity of the body can be obtained from the relation, v v ax (8ms )(1m ) 16m s 1 v 4ms The body will move with this velocity for 5s. Therefore, the distance covered with constant speed = (4ms -1 )5s = m. Exercise 4-1: The moon is m from the earth and has a mass of kg. Find the gravitational force it exerts on a 1kg body on earth; express your result also as a fraction of body s weight. The gravitational force is m1m G r 11 G Nm kg, m kg, m 11 Here, Nm kg kg1 kg Fg m kg, r.381 m In the earth the body s weight is W 1kg 9.8 ms 9. 8N Therefore, the fraction is N 6 fraction N Exercise 4-17: The mass of the moon is about one eighty-first, and its radius onefourth, that of the earth. What is the acceleration due to gravity on the surface of the moon? The acceleration due to gravity can be defined from the relation, Gm g G Here, Nm kg g kg 637m Nm kg, m, kg ms 637m 4 5 N PHY 11/Chapter-3 5

The Concept of Force. field forces d) The gravitational force of attraction between two objects. f) Force a bar magnet exerts on a piece of iron.

The Concept of Force. field forces d) The gravitational force of attraction between two objects. f) Force a bar magnet exerts on a piece of iron. Lecture 3 The Laws of Motion OUTLINE 5.1 The Concept of Force 5.2 Newton s First Law and Inertial Frames 5.3 Mass 5.4 Newton s Second Law 5.5 The Gravitational Force and Weight 5.6 Newton s Third Law 5.8