Blocks on Moving Surfaces

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Picture a table set for dinner, with plates on a tablecloth. Someone pulls the cloth out with one sharp jerk, and the plates stay on the table! But pull it slowly, and everything comes along with the cloth and crashes down. So why does a fast pull leave the plates behind? Today: a box on a truck, a block on a moving belt, and a dish on a tablecloth. Questions like these come up often in JEE, and one simple idea solves them all. So, let's begin.

0:37A box on the floor of a truck

Alright, a truck first. There is a box on its floor, and the truck starts to speed up. The box speeds up with it. But who pushes the box forwards? Nobody is touching it! Friction, yes! The floor is level, so the only horizontal force on the box is friction from the floor. The floor tries to slide forwards under the box, and friction, opposing that slipping, drags the box forwards. With me? To move with the truck, the box needs the same acceleration a as the truck. With small m as its mass, that needs a force m times a. The box is not sliding, so this is static friction, and it is at most mu s times N, right? Mu is this Greek letter, the coefficient of friction, and the small s below it means static. Capital N is the normal reaction; on a level floor it equals m g, the weight, where g is the acceleration due to gravity. So m a is less than or equal to mu s m g; that is this sign, less than with a line under it. Okay? Cancel m from both sides. So the box moves with the truck only while a is at most mu s g. And look, the mass has cancelled! Light or heavy, a box slips at the same acceleration, for the same mu s. Clear?

1:54When the truck speeds up too fast

Now, what if the truck's acceleration is more than mu s g? That is this greater than sign. Static friction cannot give that much force, so the box slips, and the friction becomes kinetic. Okay? The box is sliding backwards on the floor. So, seen from the ground, it accelerates backwards, right? No, no, careful! It slides backwards relative to the floor, so kinetic friction on it acts forwards, against the slipping. So, from the ground, the box still speeds up forwards, at mu k g. The small k means kinetic, and the small word box below a labels the box. Got it? But the truck speeds up faster than mu k g. So the truck gains on the box, and relative to the floor, the box slips backwards, towards the rear. Forwards from the ground, backwards relative to the floor. Okay?

2:46A block on a moving belt

Now, the second situation. Think of an airport conveyor belt, moving steadily at a speed v, and you place a block on it gently, like a bag. So at first, the block is at rest. Right? Now tell me. Does the block move at the belt's speed at once? No, it does not. For a while, the belt slides under the block, and friction drags the block forwards. That friction is mu m g; here the course writes just mu, with no small letter below it. So the block's acceleration is mu g. With me? Its speed grows at mu g until it equals v. It starts from rest, so speed equals acceleration times time, so this takes a time t equals v over mu g. After that, no slipping: the block just rides along with the belt. Okay? But all that time, the belt was faster than the block. So the belt runs ahead, and relative to the belt, the block slips back by v squared over two mu g. The little two up on v means squared. Clear?

3:47Back to the tablecloth

Now, let's settle our puzzle. Look, a dish on a tablecloth is just like the box on the truck, isn't it? So what decides whether the dish goes with the cloth? The limit of friction! The only horizontal force on the dish is friction from the cloth, at most mu m g, where m is now the mass of the dish. So the dish's acceleration, a with the small word dish below it, can never be more than mu g, however hard you pull. Okay? So pull the cloth with an acceleration more than mu g. The dish cannot keep up, the cloth slides out from under it, and the dish stays behind. That sharp jerk is exactly a large acceleration. Right? And a slow, gentle pull means a small acceleration, at most mu g. Then friction is enough, the dish moves with the cloth, and crash! No magic at all, just the limit of friction. Got it?

4:40Example 1: greatest truck acceleration with no slipping

Alright, example one, an easy one. Read it with me: a box lies on the floor of a truck, and mu s is zero point three. Find the greatest acceleration of the truck for which the box does not slip. Pause the video, and try it first. Done? Okay, let's check. But wait, the mass of the box is not given. Can we still solve it? Yes, we can! Friction can give the box an acceleration of at most mu s g. The mass cancels, so we do not need it. Right? Take g as ten metres per second squared; on screen, m s to the power minus two. So a max, with the small word max below a, equals mu s g: zero point three times ten. With me? Three metres per second squared. Up to that, the box rides along with the truck. Any harder, and it slips backwards on the floor. Perfect!

5:39Example 2: how long and how far a block slips on a belt

Now a harder one. Read it with me: a block is placed gently on a belt moving at three metres per second; on screen, m s to the power minus one. And mu is zero point three. Find the time for which it slips, how far it moves relative to the ground in that time, and how far it slips on the belt. Pause the video and try it. Done? Okay, let's check. Step one: friction gives the block an acceleration mu g. Zero point three times ten: three metres per second squared. Right? Step two: it slips until it reaches the belt's speed. So t equals v over mu g, three over three: one second. The small s is for seconds. Okay? Step three: the block's distance, seen from the ground. It starts from rest, so the distance is half times a times t squared. Half times three times one squared: one point five metres. With me? The belt moves steadily, so in that one second it covers three times one: three metres. Got it? The block moved one point five metres forwards. So it slips one point five metres forwards on the belt, right? Oh no, careful! The belt went three metres, the block only one point five. So the belt ran ahead, and the block fell behind on it by three minus one point five: one point five metres, backwards. Forwards relative to the ground, backwards relative to the belt. Clear? Let's check with our formula, v squared over two mu g. Three squared is nine; two times zero point three times ten is six; nine over six is one point five metres. It matches! So: one second, one point five metres, and one point five metres backwards. Very good!

7:34Quick revision: truck, conveyor belt and tablecloth

So, quick revision. Truck: the box moves with it only while the acceleration is at most mu s g. Belt: the block speeds up at mu g, and slips back by v squared over two mu g. Tablecloth: pull with an acceleration more than mu g, and the dish stays behind. Now go and try the practice problems on this page. All the best!