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Notes On Force And Motion

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Published in: Physics
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Notes on Physics for middle school students

Aritra D / Kolkata

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  1. Force and Motion In the lesson on Matter, we learnt that inside the atom, the electrons move about in orbits around the central nucleus. Similarly, the planets of the solar system move about in orbits around the Sun just as the Satellites move around the planets. In these movements, it may be noted that neither the electrons nor the planets and satellites either fall into the nucleus, the sun or the corresponding planet or wander off from their orbital paths by themselves. They keep on moving around in their fixed paths until an external source intervenes. Have you wondered why this happens? It is simply because of two equal and opposite forces acting on the electron, the planet and the satellite that keeps them in their path: the centripetal and the centrifugal force. Before we proceed to learn about the Centripetal and the Centrifugal forces, it is important to understand the basic concept of 'force'. What is Force? While playing football, you have to kick it hard either to get it into motion or to change its direction. Similarly, in cricket, the batsman has to either flick the ball to change its direction or to block it completely to prevent it from reaching the stumps. In all the cases mentioned above, force is being applied either to change the condition of 'being' of the object or to change its direction. This 'force' is an external condition and is not generated internally by the body. Similarly, before sitting down to have your meal, you pull out the chair from the side of the dining table. Before pulling out that chair, it was fixed at (stationary or static) one point and did not move from the fixed point by itself. It was the external force that you applied through your hand that caused the chair to change its position. Thus, force is the application of any external influence (a push or a pull) that changes the existing state of a body. By the application of force, a body may start moving or come to a stop or change the direction in which it was moving. Also, application of force may cause temporary or permanent shape change but the material remains the same. For example, if you stretch a rubber band, it undergoes a shape change temporarily and soon returns to its original shape after that stretching force is removed. However, if you stretch it beyond a certain point, its shape will change permanently and it will not return to its original shape. Stretching it even further will lead the rubber band to snap. Whatever is the outcome of the stretching force on the rubber band, its material will remain the same; it will not be transformed into platinum or gold or silver. There are two very interesting features about force. Firstly, it does not change the mass of the body on which the force is applied. Thus, the mass of the football or the cricket ball or the rubber band remains the same even after application of force upon them. Secondly, force is just like the air, it cannot be seen but it can definitely be felt. Force is a vector quantity because it is not only has direction but is also measurable.
  2. The Unit of Force is called Newton. IN is the force that would accelerate a 1 kg mass from rest to 1 m/s in 1 s. Thus, IN = 1 kg x 1 m/s 2 Here, accelerate means to increase the speed of a moving body with respect to time. The opposite of acceleration is decceleration or decrease of the speed of a moving body with respect to time. In Physics, acceleration, a, is the amount by which your speed changes in a given amount of time, or a = Av/At where Av is change of velocity and At is change of time. Given the initial and final velocities, vo and Vf, and initial and final times over which your speed changes, to and tf, you can also write the equation like this: a = Av/At = (Vf — to) Acceleration, like velocity is actually a vector and is often written as 'a', in vector style. In other words, acceleration, like velocity and unlike speed, has a direction associated with it. In terms of units, the equation looks like this: a = (Vf — — to) = distance/time/time = distance/time2 Momentum Force should not be confused with momentum as they are completely different concepts. While force is the product of the inertia and rate of change of velocity, momentum is the product of the inertia and the velocity of a body travelling in a straight line. Catching a runaway car going down a steep hill is a problem because of its momentum. If a car without any brakes is speeding toward you at 40 kilometres per hour, it may not be a great idea to try to stop it simply by standing in its way and holding out your hand. The car has a lot of momentum, and bringing it to a stop requires plenty of effort. The greater the mass of a body, the more momentum it has. The more the velocity, greater is the momentum. The symbol for momentum is p, so you can say that, p = mv, where m = mass of the body and v = velocity of a body in a straight line. Momentum is a vector quantity, meaning that it has a magnitude and a direction, and the magnitude is in the same direction as the velocity — all you have to do to get the momentum of an object is to multiply its mass by its velocity. Because you multiply mass by velocity, the units for momentum are kilograms-meters per second, kg-m/s, in the MKS system and grams-centimeters per second in the CGS system.
  3. Angular Momentum Imagine a 40-ton satellite rotating in orbit around the earth. You may want to stop it to perform some maintenance work, but when the time comes to grab it, you stop and consider the situation. It takes a lot of effort to stop that spinning satellite. Why? Because it has angular momentum. What then is Angular Momentum? In the previous section we learnt that linear momentum, p, equals the product of mass and velocity: p = mv Angular Momentum, L, has as little to do with the word "momentum" as the letter p does. The equation for angular momentum is L = lw where I is the moment of inertia and w is the angular velocity. Angular Momentum is a vector quantity, meaning it has a magnitude and a direction, which points in the same direction as the w vector (that is, in the direction the thumb of your right hand points when you wrap your fingers around in the direction the object is turning). The units of angular momentum are I multiplied by w, or kg-m2/s in the MKS system. Like Linear Momentum, Angular Momentum is also conserved. The principle of conservation of angular momentum states that angular momentum is conserved if there are no net torques involved. This principle comes in handy in all sorts of problems, often where you least expect it. You may come across more obvious cases, like when two ice skaters start off holding each other close while spinning but then end up at arm's length. Given their initial angular speed, you can find their final angular speed, because angular momentum is conserved, which tells you that the following is true: 1101 = 1202 If you can find the initial moment of inertia and the final moment of inertia, you're set. But you also come across less obvious cases where the principle of conservation of angular momentum helps out. For example, satellites don't have to travel in circular orbits; they can travel in ellipses. And when they do, the calculations can get a lot more complicated. The principle of conservation of angular momentum can make the problems simple. Conservation of Momentum
  4. The principle of conservation of momentum states that when you have an isolated system with no external forces, the initial total momentum of objects before a collision, equals the final total momentum of the objects after the collision (Pf = pJ. You may have a hard time dealing with the physics of impulses because of the short time spans and the irregular forces of action. The absence of complicated external forces is what you need to get a truly useful principle. Troublesome items that are hard to measure — the force and time involved in an impulse — are out of the equation altogether. For example, say that two careless space pilots are zooming towards the scene of an interplanetary crime. In their eagerness to get to the scene first, they collide. During the collision, the average force exerted on the first ship by the second ship is F12. By the impulse-momentum theorem, you know the following for the first ship: F12 At = API = mlAv1 = ml(Vf1 — vol) And if the average force exerted on the second ship by the first ship is Fm, you also know that hi At = AP2 = m2Av2 = m2(Vf2 — v02) (ii) Now you add these two equations together, which gives you the resulting equation F12At + F21At = API + AP2 = ml(Vf1 — vol) + m2(Vf2 — v02) (iii) Rearrange the terms on the right until you get F12At + F21At = (mlVf1 + m2Vf2) - (mlV01 + m2V02) (iv) This is an interesting result, because rmvol + nuv02 is the initial total momentum of the two rocket ships, + P20, and nuvfl + m2iv/f2 is the final total momentum, Pif + P2f, of the two rocket ships. Therefore, you can write this equation as follows: F12At + F21At = (mlVf1 + m2Vf2) - (nuvol + rmv02) = (Pif + P2f) - (Plo + P20) (v) If you write the initial total momentum as Pf and the final total momentum as PO, the equation becomes F12At + F21At = (mlVf1 + m2Vf2) - (r-mvol + nuv02) = Pf - PO (vi) Where do you go from here? You add the two forces together, F12 At + hi, to .get the sum of the forces involved, EF ZFAt = Pf- PO ... (vii)
  5. If you're working with what's called an isolated or closed system, you have no external forces to deal with. Such is the case in space. If two rocket ships collide in space, there are no external forces that matter, which means that by Newton's third law, F12 = — hi. In other words, when you have a closed system, you get (viii) This converts to ... (ix) The equation Pf = po says that when you have an isolated system with no external forces, the initial total momentum before a collision equals the final total momentum after a collision, thus giving you the principle of conservation of momentum. Now, we will look into another significant aspect of Conservation of Momentum. When two objects collide and stick together, is the final total momentum really equal to the initial total momentum? Well, not exactly. If one object rams another and the two stick together, a lot of friction could be involved, which means that some kinetic energy gets converted into heat energy. When that happens, the final momentum may not exactly be equal to the initial momentum. Sometimes, because of impacts the objects become deformed, as when two cars crash and produce dents. In that case, too, some energy is used in deforming the materials, so the final momentum may not equal the initial momentum. Friction and its role shall be discussed in greater detail in subsequent classes. Impulse In terms of physics, impulse tells you how much the momentum of an object will change when a force is applied for a certain amount of time. For example, when you are playing carom, you hit the coins on the board with the striker. The force with which the striker hits a coin and the product of the duration of the impact between the striker and the coin gives us the impulse of the object. Thus, Impulse = FAt where At is the duration of time = m x a x At where m = mass of the body and a = acceleration of the body = m x Av/At x At where Av/At = (Vf — to = m x Av Hence, Impulse of a body can also be defined as the change in momentum of a body. Impulse is a vector, and it's in the same direction as the average force. You get impulse by multiplying Newtons by seconds, so the units of impulse are Newton-seconds or kg m/s in the meter-kilogram-second (MKS) system, dyne-seconds or g cm/s in the centimeter-gram-second (CGS) system, and pound-seconds in the pound-foot-second system.
  6. Torque Torque is a measure of the tendency of a force to cause rotation. The torque exerted on an object depends on where you exert the force. You go from the strictly linear idea of force as something that acts in a straight line, such as pushing a refrigerator up a ramp, to its angular counterpart, torque. Torque brings forces into the rotational world. Most objects aren't just point or rigid masses, so if you push them, they not only move but also turn. For example, if you apply a force tangentially to a merry-go-round, you don't move the merry-go-round away from its current location — you cause it to start spinning. How much torque you exert on an object depends on the point where you apply the force. The force you exert, F, is important, but you can't discount the lever arm — also called the moment arm — which is the distance from the pivot point at which you exert your force. Let us assume that you're trying to open a door. You know that if you push on the hinge the door won't open; if you push the middle of the door, it will open slowly; and if you push the edge of the door, the door will open faster. The lever arm, l, is the distance from the hinge at which you exert your force. The torque is the product of the force multiplied by the lever arm. It has a special symbol, the Greek letter -c (tau): t = Fl The units of torque are force multiplied by distance, which is Newtons-meters in the MKS system, dynes-centimeters in the CGS system, and feet-pounds in the foot-pound-second system. So, for example, the lever arm is at a distance r, then = Fr. If you push with a force of 200N, and r = 0.5 meters, what's the torque? If you push at the hinge, so your distance from the pivot point is zero, which means the lever arm is zero. Therefore, the torque is zero. If you exert the 200N of force at a distance of 0.5 meters perpendicular to the hinge, then t = Fl = 200 x (0.5) = IOON-m The torque here is 100 N-m. Again, if you push with 200N of force at a distance of 2r perpendicular to the hinge, which makes the lever arm 2r = 1.0 meter, so you get torque t = Fl = 200 x (1.0) = 200N-m Kinds of Forces Forces are broadly categorised into two kinds of forces: Contact Forces and Distance Forces or Forces at a Distance. Contact forces are those forces where there is physical contact between the
  7. object on which the force is being applied and the agent of the application of force. In all the above applications, the contact forces were acting on the football, the cricket ball, the rubber band and the chair. Some other examples of contact forces are: frictional forces, and mechanical forces like tension and torque. Some examples of forces applied at a distance include electrostatic force, electromagnetic force, gravitational force, centripetal force, centrifugal force, Van der Waal's force etc. All these forces influence the movements (motion) of bodies starting right from the subatomic particles up to the galactic bodies. Motion Motion is the action of changing location or position. The study of motion without regard to the forces or energies that may be involved is called kinematics. It is the simplest branch of mechanics. The branch of mechanics that deals with both motion and forces together is called dynamics and the study of forces in the absence of changes in motion or energy is called statics. The term energy refers an abstract physical quantity that is not easily perceived by humans. It can exist in many forms simultaneously and only acquires meaning through calculation. A system possesses energy if it has the ability to do work. The energy of motion is called kinetic energy whenever a system is affected by an outside agent, its total energy changes. In general, a force is anything that causes a change (like a change in energy or motion or shape). When a force causes a change in the energy of a system, physicists say that work has been done. The mathematical statement that relates forces to changes in energy is called the work-energy theorem. When the total of all the different forms of energy is determined, we find that it remains constant in systems that are isolated from their surroundings. This statement is known as the law of conservation of energy and is one of the really big concepts in all of physics, not just mechanics. There are basically four types of motion: 1. Translational motion results in a change of location along a straight line. 2. Oscillatory motion is repetitive and fluctuates between two locations. This second type of motion is seen in pendulums (like those found in grandfather clocks), vibrating strings (a guitar string moves but goes nowhere), and drawers (open, close, open, close — all that motion and nothing to show for it). Oscillatory motion is interesting in that it often takes a fixed amount of time for an oscillation to occur. This kind of motion is said to be periodic and the time for one complete oscillation (or one cycle) is called a period. Periodic motion is important in the study of sound, light, and other waves.
  8. 3. Rotational motion occurs when an object spins. Rotational motion too is often periodic. 4. Random motion occurs for one of two reasons — (a) Some motion is predictable in theory but unpredictable in practice, which makes it appear random. For example, a single molecule in a gas will move freely until it strikes another molecule or one of the walls containing it. The direction the molecule travels after a collision like this is completely predictable according to current theories of classical mechanics. (b) Some motion is unpredictable in theory and is truly random. For example, the motion of the electron in an atom is fundamentally unpredictable. The harder you try to locate the electron, the less you know about its velocity. The harder you try to measure its velocity, the less you know about its location. This is fundamental quality of small objects like electrons and there is no way around it. Newton's Laws of Motion Sir Isaac Newton made three seminal conclusions about motion and forces that have stood the test of time and have served the basics of Physics ever since. These three together are known as Newton's Laws of Motion. Let us try to understand these Laws one by one. First Law of Motion: An object initially at rest is predicted to remain at rest if the total force on it is zero, and an object in motion in a straight line remains in motion with the same velocity in the same direction. The converse of Newton's first law is also true: if we observe an object moving with constant velocity along a straight line, then the total force on it must be zero. This means that the body experiences zero acceleration as there is no change of velocity with respect to time. However, this is a hypothetical situation as it cannot be that a body is perpetually at rest or perpetually in motion. Let us examine why. Suppose a ball is rolling on a plane floor and no external force is applied to it to either change its velocity or stop it. It will still come to rest because of the frictional force that comes into play and this frictional force shall cause its velocity to reduce. Similarly, the ball remain stationary on the plane floor remains so because of the force of friction acting between the surfaces of the floor and the ball. If there was no friction, then the ball would not have stood still. In reality, a frictionless surface does not exist. Second Law of Motion: The net force acting on a body is equal to the product of the body's mass and its acceleration. This may be mathematically represented by the equation, F = M x where F is the direction of application of Force on a body; M is the mass of the body; and
  9. is the acceleration or the rate of change of velocity (direction of rate of change of speed) Example: A bus with a mass of 2000 kg accelerates from 0 to 25 m/s in 34 s. Assuming that the acceleration is constant, what is the total force on the bus? Solution: We solve by using the equation of Newton's second law. Thus, F = M x a, = (2000 kg) x (25 m/s - 0 m/s)/(34 s) = 2000 x 25/34 = 1470.6 N Third Law of Motion: Forces occur in equal and opposite pairs: whenever object A exerts a force on object B, object B must also be exerting a force on object A. The two forces are equal in magnitude and opposite in direction. Thus, every action has an equal and opposite reaction. Thus, when a gun is fired, the bullet moves out of because of the force exerted on it by the hammer that pushes it. At the same time, the bullet also exerts an equal and opposite force on the hammer that has a recoil effect on the gun. Similarly, a rocket or space ship takes off because of the equal and opposite force exerted on it by the recoil of the fuel that is discharged from it with a large force. Inertia This term is closely related to Newton's first and Second Laws. It is the quality of an object by which, (a) a body at rest continues to stay at rest or (b) a body in motion in a straight line continues to stay in motion in the straight line until an external force is applied to it to overcome that state of rest or of motion. Inertia is a quality of mass, and the mass of an object is really just a measurement of its inertia. Inertia, the tendency of mass to preserve its present state of motion, can be a problem at times. Refrigerated meat trucks, for example, have large amounts of frozen meat hanging from their ceilings, and when the drivers of the trucks begin turning corners, they create a pendulum motion they can't stop in the driver's seat. Trucks with inexperienced drivers can end up tipping over because of the inertia of the swinging frozen load in the back. Because mass has inertia, it resists changing its motion, which is why you have to start applying forces to get velocity and acceleration. Mass binds force and acceleration together. Equilibrium
  10. In physics, an object is in equilibrium when it has zero acceleration — when the net forces acting on it are zero. The object doesn't actually have to be at rest — it can be going 1,000 miles per hour as long as the net force on it is zero and it isn't accelerating. Forces may be acting on the object, but they all add up, as vectors, to zero. Centripetal and Centrifugal Forces Having learnt up all the basic concepts, it is time we returned to the two kinds of forces mentioned at the beginning of this chapter: Centripetal and Centrifugal Forces. The Centripetal Force is the force that is directed towards the centre of a circle, which keeps an object moving in a circular motion. We learnt that force equals mass times acceleration, F = ma. In case of turning forces, centripetal acceleration = v2/r where r is the radius of the circular field of motion. Now you can determine the centripetal force needed to keep an object moving in uniform circular motion with mass m, speed v, and radius r with the following equation: FC = mv2/r This equation tells you how much force you need to move a given object in a circle at a given radius and speed. Objects moving in circles with the same radius can have different speeds, and so more force is needed to move objects at faster speeds. For example, a ball is moving at 12.6 m/s per second on a circular string of radius 1 m. Calculate how much force is needed to make a 10 kg cannonball move in the same circle at the same speed? Given, mass of cannon-ball = 10 kg; speed of cannon-ball = speed of ball = 12.6 m/s; radius of circular string = 1 m 2 = 1,590 Newtons = mv /r = Centripetal force isn't some new force that appears out of nowhere when an object travels in a circle; it's the force the object needs to keep traveling in that circle. For that reason, it may be better to refer to it as centripetal-needed force. Centripetal force is the force needed to keep the object going in a circle — and to keep it going, the vector sum of all the other forces (frictional, gravitational etc.) on that object gives the centripetal force needed. Imagine that you're driving a car, riding a bicycle or jogging on a surface and you come to a banked curve. The steeper the bank of the surface, the more centripetal force you need to steer a car, a bicycle or yourself around it at a high speed. The force you need comes from the friction of the tires with the surface or from the friction of your shoes against the ground; if the surface is covered with a foreign substance such as ice, you produce less friction, and you can't turn as safely at high speeds.
  11. Centrifugal Force is the apparent force that is dealt by an object moving in a curved path that acts outwardly away from the centre of rotation. We feel this force when we go round a corner in a car or an aeroplane banks into a turn. We see this also in the spin cycle of a washing machine, a merry-go-round or even a mixer used in the kitchen. It is the same force that generates a feeling of being 'thrown away'. Thus, while twirling a string with a stone tied to its end, the string exerts an inward force to try and come back to the centre while the stone exerts an outward force to escape. Both forces balance each other to create that circular motion. Similarly, these are the two counterbalancing forces that keep electrons, planets and satellites revolving in their orbits—neither letting them escape from their circular paths nor letting them collapse into the centre of the nucleus, the star or the planet.

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