The Common Forces

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Take a spring balance, and tie a string to each end. Pass each string over a fixed pulley, and hang a five kilogram mass from each one. Each mass weighs fifty newtons. So, what does the balance read: fifty newtons, or one hundred? Most students say one hundred, straight away. Hold that thought. We will settle it at the end. To answer it, we need the common forces of mechanics: weight, the normal force, friction, tension and the spring force. You will use them in nearly every JEE mechanics problem. So, let's begin.

0:44Weight and the normal force

So, first, the force you already know: weight. W equals m g. Capital W is the weight, m is the mass, and g is the acceleration due to gravity, okay? It acts vertically down, at the centre of gravity of the body. Right? Now, a question for you. Put a book on a table. Gravity pulls it down all the time. So why doesn't it fall through the table? Yes, exactly! The table pushes it up. That push is the normal force, written as capital N, in italics. Normal means perpendicular. The push of a surface is always perpendicular to the surface, okay? For the book on the table, the normal force comes out equal to m g. So tell me: is N always equal to m g? No, no, careful! This is a classic trap. The normal force has no fixed formula. It adjusts itself, just enough to stop the two bodies from going into each other. Interpenetrating is the long word for that. Push down on the book, and the table pushes back harder. With me? And when the two bodies separate, the normal force becomes zero. So, N equals zero is the moment contact is lost. Remember that; it is very useful.

1:59Friction and the total contact force

Clear? Now, the normal force is only one part of what a surface does. The part of the contact force along the surface is friction, small f. So friction acts along the surface, while N acts perpendicular to it. So the surface gives two pushes at once: N, perpendicular to it, and f, along it. These two are at right angles. So how do we combine them into one total force? Like the two sides of a right triangle, yes! The total contact force is the square root of, N squared plus f squared. The tick-shaped sign is the square root, and the small two up top means squared, okay? So if a question asks for the total contact force, do not stop at N alone. Got it?

2:45Tension: the pull of a string

Next, tension. When a string tied to a body is pulled tight, it pulls the body. That pull is the tension, capital T, okay? It acts along the string, and away from the body. Right? Now, an important condition. If the string is light, the word in brackets, massless, then the tension is the same all along it. And it stays the same even when the string passes over a pulley, as long as the pulley is light and frictionless. Light string, light frictionless pulley: keep both conditions in mind. Clear?

3:20Spring force: Hooke's law

Now the spring force. Hooke's law, written in brackets: F equals k x. Here k is the spring constant, and x is the extension or the compression: how far the spring is stretched or squeezed, okay? Double the stretch, and the force doubles too. Now, the direction of this force. The spring always tries to restore its natural length. Stretched, it pulls back in. Compressed, it pushes back out. Got it?

3:47Combining springs

Alright, now two springs together. Join two springs end to end, and pull the pair. Is the pair stiffer than each spring, or softer? Softer, yes! This is series. Both springs carry the same force, written in brackets, and their stretches add up. So, one over k equals one over k one, plus one over k two. k one and k two, with the small one and two written below, are the two spring constants, okay? Careful: in series, do not simply add the k values. That is the usual mistake. Now, put the two springs side by side. That is parallel. Here both have the same extension, and their forces add. So k equals k one plus k two. Simple addition this time. Right? Here is a topper's trick. Cut a spring of constant k into n equal pieces, and each piece has constant n k. Shorter spring, stiffer spring. With me? And the last line, which settles our puzzle. Any spring balance reads the tension in it, not the sum of the forces at its two ends. Remember this one. We will use it in the second example. Clear?

4:59Example 1: spring force and stretch with a hanging block

Now, Example one. Here, a spring of constant two hundred newtons per metre is stretched by five centimetres. On screen, the upright capital N is the unit newton, not the normal force, and m to the minus one means per metre, okay? Find the force it exerts. Then, how much does it stretch when a two kilogram block hangs from it, at rest? Pause the video, and try it first. Done? Okay, let's check. F equals k x. So, two hundred times five, the cross on screen meaning times: one thousand newtons, right? Oh no, careful! The spring constant is in newtons per metre, so x must be in metres too. Five centimetres is zero point zero five metres. Now F equals k x: two hundred times zero point zero five. That gives ten newtons. That is the force the spring exerts. Next, the block. It hangs at rest, so the spring force balances its weight: k x equals m g. With g as ten metres per second squared, m g is two times ten: twenty newtons. Clear? So x equals twenty over two hundred, that is zero point one metre. Ten centimetres. Perfect!

6:16Example 2: spring balance with 5 kg at each end

Now, Example two, and this is our puzzle from the start. Here, a spring balance has a five kilogram mass hanging from each end, with the strings passing over two fixed pulleys. What does it read? Pause the video, and try it. Done? Okay, let's check. Take one mass first. It hangs at rest, so the tension in its string balances its weight. T equals m g: five times ten, fifty newtons. Now, take the strings as light, and the pulleys as light and frictionless. Then the tension stays fifty newtons all along each string, okay? So each end of the balance is pulled with fifty newtons. Fifty plus fifty: the balance reads one hundred newtons, right? No, no, wait! That is the trap. Fix one end of the balance to a wall, and pull the other end with fifty newtons. The wall pulls back with fifty newtons too, and everyone agrees the balance reads fifty. Our setup is exactly the same. The second mass simply does the wall's job. So, the reading equals the tension: fifty newtons. Not one hundred, as the brackets remind you. Puzzle solved! Very good!

7:32Quick revision: weight, normal, tension and springs

So, quick revision. Weight, m g, acts down. The normal force is perpendicular to the surface and adjusts itself. Friction acts along the surface. In a light string, the tension is the same all along. And a stretched or compressed spring exerts a force k x. In series, add one over k. In parallel, add k. And a spring balance reads the tension, not the sum. Now go and try the practice problems on this page. You are ready for them. All the best!