Triple Physics Topic 6 - Centre of Mass

Centre of Mass

An important principle in Physics is that any object, however sophisticated, large, or complex can be treated as a point mass.  When we work out the effect of forces, we show the forces all acting at a single point.  This point is called the centre of mass.

 

We treat objects as point masses referring to a single point called the centre of mass.

In regular objects like a cube or a sphere, the centre of mass is in the middle.  In some objects the centre of mass is outside the object.

The centre of mass (or centre of gravity) is the point through which the entire weight is said to act.  Objects with a very low centre of gravity tend to be very stable.  Some objects are so stable that they never fall over.  Objects with a high centre of gravity are unstable.

 

Finding the centre of mass for irregular objects

We can find the centre of mass of an irregular object quite easily.  If we let it hang freely, the centre of mass is directly below where we hang it from.  In old text books, an object like this is called an irregular lamina.  (The word lamina is a Latin word for leaf.)

 

 

We draw a line vertically downwards.

 

If we then hang the object from a couple of other points and draw the lines that go vertically downwards, the centre of mass is where the lines meet.

 

When the object is hanging freely, the centre of mass is vertically below the hanging point.  The vertical arrow is called the line of action of the weight.

 

 

 

 

Centre of Mass and Stability

A stable object does not tip over.  Objects that have a high stability do not tip over easily.  Their centre of mass is low down, near the base.  This candlestick has a low centre of mass so that it does not tip over so easily.

 

 The candlestick is in stable equilibrium.  If you push the top, it will drop back to where it was.

 

 

Now suppose we put it upside down.  This time the centre of mass is high up.  It is in unstable equilibrium and if you pushed the candlestick, it would tip over easily.

 

 

Now we put it on its side:

 

 

If you push it, it will roll, but will not tip over.  It is in neutral equilibrium.

 

This wine rack uses the idea of moments to form a stable object.

 

 

 

Question 1

Look at the picture above.

(a) Where is the centre of mass in the bottle?

(b) Where is the pivot?

(c) Where does the line of action from the centre of mass of the bottle act?

(d) Use the principle of moments to explain how this system balances.

Answer

 

 

 

Centre of Mass and Stability

Let us look at how we can explain stability in vehicles.  This bus has a low centre of mass and a wide track (distance between the wheels)

 

You can see that there is a line of action of the weight that acts vertically downwards from the centre of mass.

 

 

 

If the bus tilts, the line of action will pull the bus upright.

 

Although they are tall, double-decker buses are very stable.  They test buses by putting lots of sandbags on the seats upstairs (with nothing downstairs) and tilt them over on a tilting platform.  The centre of mass is low enough to ensure that they are tilted to more than 60o off the vertical before they tip over.

 

Lorries have a higher centre of mass on their trailers, due to the load.  If you live in the country and get stuck behind a hay-lorry, you may see it swaying alarmingly. This kind of accident tends to happen with lorries when they drive through strong cross-winds on exposed roads.

 

Question 2

Explain two measures that automotive engineers take to ensure that vehicles are stable.

Answer

 

Yachts have a deep keel to make the centre of mass a low as possible, so that if they get tipped on their side, they are pulled upright by a the turning effect of the centre of mass.  They are described as self-righting.

 

A ship with a high centre of mass can tip over easily.  Henry VIII's battleship, Mary-Rose, did just that.

 

 

 

Pendulum

A pendulum is a simple system consisting of a mass (often called a bob) that hangs freely below a fixed point on a thread.  Normally the mass will hang vertically below the fixed point.  That is the idea you used when working out the centre of mass of an irregular object.

  

It doesn't matter what shape the bob is.  We treat it as a single point mass and the thread is light and does not stretch.  If we push the bob to one side, we find that the pendulum will swing from side to side.  It oscillates.  Each oscillation (complete to-and-fro movement) has a time period.

 

 

Question 3

Explain how energy is interchanged between two different forms in a pendulum.

Answer

 

Period depends on the length of the thread.  It does NOT depend on the mass, or the material, of the bob.  As long as the angle of swing is small, the angle does not affect the periodicity of the swing either.  However, if we swing through a large angle, the period is different (by a small amount) from the period of a small swing.  The periodicity is important, as it enables pendulum clocks to measure time with good accuracy.  Many clocks in churches and other public buildings have their time-keeping governed by the swing of a pendulum.

 

A child's swing works like a pendulum.

 

 

If no energy is put into the swing, the child will eventually stop swinging.  This is because a certain amount of energy is lost through friction.  You may remember that some swings squeak because they have not been oiled.  The swing will stop quite quickly.  We call this reduction in swing damping.

 

 

Period and Frequency

We measure the time period of the pendulum by timing 10 (or more) complete swings.  We then divide by 10 (or whatever) to get the time taken by one swing.  This time is called the period, and is given the Physics code T.  Frequency is the number of oscillations per second, and has the Physics code f.  They are related by a simple formula:

 

Frequency = __1___

                 Period

 

In Physics code:

f = 1

     T

Frequency is measured in Hertz (Hz).  Period is measured in seconds (s).

 

Question 4

It takes 36 seconds for a pendulum to do 20 oscillations.  What is the period?  What is the frequency?

Answer

 

The period of the swing of a pendulum depends on the square root of its length.  This means that to get double the period, you need four times the length.  You are NOT expected to use the pendulum formula, which you will meet in Unit 4 of the Physics A-level course.

 

Periodic oscillations are studied in more depth at A-level, when we study simple harmonic motion.