Composite Bodies and Bodies with Cavities

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle before we start. Take a uniform disc, a flat round plate. Its centre of mass is right at the centre. Now I cut a small circular hole near the right edge. So tell me: does the centre of mass move towards the hole, or away from it? And by how much? Today we settle it in one line, using a strange idea: negative mass! Such problems come up often in JEE, so let's begin.

0:30Composite bodies: one point mass per part

So, first, composite bodies: bodies built from simple pieces, like rods, plates or solids. To find the centre of mass of such a body, must we integrate over the whole thing? No, not at all! Replace each part by a single point mass, placed at that part's own centre of mass. The arrow on screen means becomes, okay? For simple pieces, you already know these points. Now the body is just a set of particles, so we use the particle formula. x c m, with the small c m below for centre of mass, equals m one x one, plus m two x two, and so on, over m one plus m two, and so on. The small one and two label the parts, and the three dots mean and so on. The same formula gives y c m. Right? But often the question gives only the sizes of the pieces, not their masses. So where do the masses come from? From the sizes themselves, yes! In a uniform body the mass is spread evenly, so the masses of the parts are in the ratio of their lengths for wires, their areas for plates, and their volumes for solids. But only for a uniform body. Clear?

1:45Cavities: the negative-mass method

Now the second idea: a body with a hole, or a cavity, cut out of it. Start with the complete body, before any cutting. Its mass is capital M, and its centre of mass is at r one. The little arrow on top of the r means a position vector, okay? Then a piece is cut away. Its mass is small m, and its centre of mass is at r two. With me? Now think. The remainder, plus the piece we cut out, gives back the full body. So what is the remainder? Yes, exactly! The full body, minus the piece. So we treat the hole as a negative mass, minus m, sitting at r two. That is the negative mass method. The particle formula then gives: r c m equals, M r one minus m r two, over, M minus m. Clear? Now, a smart choice of origin: the centre of mass of the full body. Which term in this formula drops out? The M r one term, correct! Because r one is now zero, as in the bracket on screen. So r c m is just minus m r two, over, M minus m. That minus sign means the centre of mass lands on the opposite side from the hole. Got it? So the remainder's centre of mass shifts by m d, over, M minus m, where d is the distance of the hole's centre from the centre of the full body. And it shifts away from the hole. There is our puzzle, solved! We removed mass from the hole's side, so the other side is now heavier. Makes sense, right?

3:20The trap: capital M is the body before cutting

Now, the trap. Look carefully. In this formula, capital M is the mass of the complete body, before cutting, not the mass that is left over. Okay? So if a question gives you the mass of the remaining body, first convert it to the full mass, then use the formula. Right? Next, how do we get small m from capital M? Take a uniform disc of radius capital R, with a hole of radius small r. How does the mass depend on the radius? Through the area, yes! For a uniform plate, mass goes with area, and area goes as the radius squared. So for a disc hole, m over M equals, r over R, in brackets, whole squared. Clear? But for a spherical cavity in a uniform solid sphere, the mass goes with the volume, and volume goes as the radius cubed. So m over M equals, r over R, whole cubed. Square for a disc, cube for a sphere. Do not mix them up! With me?

4:20Example 1: two rods in a T, centre of mass from junction

Alright, Example one. Two identical uniform rods, each of length L, form the letter T, as written in the question. One is horizontal, and the other hangs vertically from its midpoint. Find the centre of mass, measured from the junction, where the rods meet. Pause the video, and try it first. Done? Okay, let's check. Composite body, so each rod becomes one point mass at its own centre. First, where is the centre of mass of the horizontal rod? At its midpoint, and that is exactly the junction! So put the origin at the junction, with y measured upwards. This point mass sits at y equals zero. Okay? The vertical rod hangs down, and its centre is at its midpoint, L over two below the junction. So its y is minus L over two; minus, because it is below. Right? Now the particle formula. The rods are identical, so each has the same mass, small m. y c m equals, m times zero, plus m times minus L over two, over two m. The m cancels: minus L over four. Clear? So the centre of mass is L over four below the junction. And sideways? Both point masses lie on the vertical line through the junction, so the centre of mass is on the vertical rod. Perfect!

5:47Example 2: solid sphere with a cavity of radius R/2

Now a harder one. From a uniform solid sphere of radius capital R, a spherical cavity of radius R over two is cut out, so that the cavity's surface passes through the centre of the sphere. Find the centre of mass of what is left. Pause and try. Hint: it is a sphere, not a disc. Done? Okay, let's check. The cavity's surface just reaches the centre of the big sphere. So how far is the cavity's centre from the sphere's centre? R over two, exactly! One radius of the cavity, from its edge at the centre to its own centre. So d equals R over two. Okay? Now, the mass removed. The radius is half, and half squared is one fourth. So small m equals M over four, right? No, no, careful! That is the disc rule. This is a sphere, so we cube the ratio. Half, in brackets, cubed, is one eighth, so small m equals M over eight. The exact trap we just saw. Got it? The remainder has mass M minus M over eight, that is seven M over eight. With me? Now the shift: m d, over, M minus m. That is, M over eight times R over two, over seven M over eight. Cancel the M and the eights: R over two, over seven. So the shift is R over fourteen. Right? So the centre of mass of the remainder is R over fourteen from the centre, on the side away from the cavity. Away from the hole, just like our disc puzzle. That's it!

7:28Quick revision: point masses, negative mass, disc vs sphere

So, quick revision. Composite body: each part becomes a point mass at its own centre of mass. Cavity: treat the hole as a negative mass, as in the formula on screen. The centre of mass shifts by m d, over, M minus m, away from the hole. And capital M is the full body before cutting: square the radius ratio for a disc, cube it for a sphere. Now go and try the practice problems on this page. You will do them well. All the best!