Centre of mass
In many applications it is important that objects are designed with stability in mind. This requires an understanding of the centre of mass, as well as an ability to find out where it is. By incorporating a low centre of mass and wide base into an object, we can reduce the chance of it toppling over.
Centre of mass 2
Mass is the amount of matter an object has. Every part of an object forms part of its overall mass. But when we try to balance an object on a point, there will only be one place where it will balance. You can therefore think of the mass of an object being concentrated at this point, known as the centre of mass.
Finding the centre of mass for symmetrical objects
The centre of mass for a symmetrical object can be found easily. The axes of symmetry are marked on the object. The centre of mass is where the axes of symmetry cross.
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Finding the centre of mass by suspending objects
The centre of mass for an irregular shaped, non-symmetrical object is found in a different way.
Drill a small hole in the object and hang it up so that it is free to swing without obstruction.
Hang a plumb line (a piece of string with a weight hanging from it) from the same suspension point. This lets you mark the vertical line directly below the suspension point.
Drill another hole at a different location within the object.
Again hang a plumb line to determine the vertical and mark it on.
The point at which the two marked lines cross is the centre of mass.
Note - you should be able to describe how to do this for your exam.
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A plumb line is an example of a simple pendulum. This is a simple machine consisting of a weight (called a bob) suspended from a suspension point by a thin piece of material such as string or a chain. The bob should be free to swing.
Common examples of pendulums include:
swings at playgrounds
some fairground rides - eg pirate ship rides
the inside mechanisms of some clocks - eg grandfather clocks
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Calculating time period for a pendulum
The time period for a pendulum, T, is the time taken for a pendulum to swing from one side to the other, and then back again to its original position.The number of complete swings (from one side to the other and back again) made by a pendulum per second is its frequency, f.
Time period and frequency are related by the equation:
T = 1/f
T = time period in seconds, s
f = frequency in Hertz, Hz
The time period of one swing of a pendulum is dependent only upon the length of the pendulum and not upon the mass of the bob, or how high it swings. Longer pendulums have greater time periods than shorter pendulums.
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Stability of objects
Stability is a measure of how likely it is for an object to topple over when pushed or moved. Stable objects are very difficult to topple over, while unstable objects topple over very easily.
The stability of an object is affected by two factors:
the width of the base of the object
the height of its centre of mass
Objects with a wide base, and a low centre of mass, are more stable than those with a narrow based and a high centre of mass.
If you are standing in a bus that is accelerating or braking, you usually spread your feet apart to increase the width of your base to make you more stable.Everyday objects are also designed with this in mind. For example, a traffic cone has a wide base and is weighted at the bottom to give it a low centre of mass.Buses have a wide base between the tyres and a low centre of mass (because the heavy engine is mounted low down).
We utilise the turning effect of forces (moments) on a daily basis, for example when we use devices such as levers. However, in some circumstances we need to prevent the turning effect of forces by balancing them with an opposing moment. Understanding the principles involved allows us to both utilise and prevent the turning effect of forces.
A moment is the turning effect of a force around a fixed point called a pivot. For example, this could be a door opening around a fixed hinge or a spanner turning around a fixed nut.The size of a moment depends on two factors:
the size of the force applied
the perpendicular distance from the pivot to the line of action of the force
This explains why less force is needed to open a door by pushing at the side furthest from the hinge than at the side closest to the hinge. To push at the hinge side of the door requires more force to be exerted because the distance is smaller.A moment can be calculated using this equation:
M = F × d
M = the moment of the force in newton-metres, Nm, F = the force in newtons, N, d = the perpendicular distance from the line of action of the force to the pivot in metres, m
Where an object is not turning around a pivot, the total clockwise moment must be exactly balanced by the total anti-clockwise moment. We say that the opposing moments are balanced:
sum of the clockwise moments = sum of the anti-clockwise moments
A see-saw has a pivot in the middle:
the person on the right exerts a force downward - which causes a clockwise moment
the person of the left exerts a force downward - which causes an anti-clockwise moment
If the people are identical weights and sit identical distances from the pivot, the see-saw will balance. This is because the total clockwise moment is balanced by the total anti-clockwise moment.The see-saw can still be made to balance even if the people are different weights. To do this, the person with the bigger weight must sit closer to the pivot. This reduces the size of the moment so the opposing moments are once again balanced.
Construction cranes lift heavy building materials using a horizontal arm called a jib. To prevent the crane toppling over, concrete blocks are suspended at the other end of the jib. They act as a counter-weight to create a moment that opposes the moment due to the load.
A lever is a simple machine that makes work easier to do. Examples of simple levers include cutting with scissors, or lifting the lid on a tin of paint with a screwdriver. Levers reduce the force needed to perform these tasks.When someone uses a lever, they exert a force (the effort) around a pivot to move an object (the load).Levers rely on the principle of moments to act as ‘force multipliers’ - they reduce the effort needed to move the load by increasing the distance over which it is acting. This means a relatively small effort force has a much greater effect.
A hammer can be used to pull out a nail from a piece of wood.The load force is 50 N and it acts at a perpendicular distance of 0.07 m. Its moment is 3.5 Nm (50 × 0.07).The effort force acts at a longer perpendicular distance. This is 0.28 m or four times the distance of the load force. As a result, the effort needed is four times less than the load force, or 50 ÷ 4 = 12.5 N.Note that the moment of the effort is 3.5 (12.5 × 0.28) – the same as the moment of the load.In this case an effort force of 12.5 N is sufficient to pull against the load force of 50 N, making it relatively easy to pull the nail out.
Calculating how to balance moments – Higher tier
Step 1: Work out the moment for which you have been given all of the information
In this case it is the anti-clockwise moment.
= force × perpendicular distance
= 500 × 2
= 1000 Nm
Step 2: Change the subject of the equation to calculate the force
Remember that for the see-saw to be balanced, the total anti-clockwise moment must be equal to the total clockwise moment. Therefore the clockwise moment must be 1000 Nm.
= force × perpendicular distance
= moment ÷ perpendicular distance
= 1000 ÷ 1.5
= 666.7 N
Stability – Higher tier
Weight pulls from an object’s centre of mass in a vertical direction toward the Earth. This is known as the line of action of the object’s weight.As an object is tilted, the line of action will continue to pull down in a vertical direction. If the line of action moves outside the base of the object, there will be a resultant moment and the object will topple over. For example, consider a lab stool.In the left-hand scenario, the line of action of the stool’s weight is acting downwards from the centre of mass in the centre of the stool’s base. Note that the centre of mass is not within a solid part of the chair.
In the middle scenario, the stool has been tipped slightly. However, the line of action of the stool’s weight is still within the base of the stool. Therefore the clockwise moment is greater than the anticlockwise moment and the stool falls back to its upright position.
In the right-hand scenario, the stool has been tipped even further. Now the line of action of the stool’s weight falls outside the base of the stool. Therefore the anti-clockwise moment is greater than the clockwise moment and the stool topples over.
Pressure can be transmitted through liquids. In hydraulic machines, exerting a small force over a small cross-sectional area can lead to pressure being transmitted, creating a large force over a large cross-sectional area. This ability to multiply the size of forces allows hydraulics to be used in many applications such as car-braking systems.
Pressure in liquids
Particles in liquids are close together, making liquids virtually incompressible. As the particles move around, they collide with other particles and with the walls of the container. The pressure in a liquid is transmitted equally in all directions, so a force exerted at one point on a liquid will be transmitted to other points in the liquid.Pressure is calculated using the equation:
P = pressure in pascals, Pa
F = force in newtons, N
A = cross-sectional area in metres squared, m2
The pressure in a liquid is equally transmitted in all directions. This means that when a force is applied to one point of the liquid, it will be transmitted to other points within the liquid.This principle can be exploited in hydraulic machines. Imagine that two syringes of different sizes were connected by tubing and filled with water.
An effort force exerted on the plunger for syringe A puts greater pressure on the water in tube A. As water is virtually incompressible, the pressure is transmitted through the water into syringe B. The water pushes against the plunger in syringe B with equal pressure, exerting a load force on it.
However, tube B has a plunger with a bigger cross-sectional area than tube A. This means that the load force exerted is larger than the effort force exerted. This is known as a force multiplier
Hydraulic systems therefore allow smaller forces to be multiplied into bigger forces. Note, however, that the bigger syringe moves a shorter distance than the smaller syringe.
Applications of hydraulics
It takes a large force to slow down or to stop a car that is travelling at speed. Hydraulics are used in the braking system of a car. They cause a relatively small force from the driver’s foot to be multiplied to produce a greater force, which acts equally on all four brake pads.The force from the driver’s foot (the effort force) exerts pressure on the brake fluid in a small piston. The pressure is transmitted throughout the brake fluid in all directions.Next to each brake disc, there is a much larger piston with a greater cross-sectional area. The transmitted pressure acts on this larger area to produce a larger load force on the brake pads. The pads then rub against the brake discs and cause the car to slow down.
Hydraulic systems are also found in:
lifting equipment - eg hydraulic jacks and wheelchair lifts
lifting and excavating arms on machinery such as diggers
hydraulic presses - which are used during the forging of metal parts
wing flaps and some rudders on aircraft and boats
Objects travelling in a circular motion are prevented from moving off in a straight line by centripetal force. This resultant force pulls objects toward the centre of the circle, continually changing the direction that an object is travelling in to keep it in circular motion.
Circular motion 2
There are many examples of objects travelling in a circular motion. For example:
a hammer-thrower spinning a hammer
the Earth orbiting the Sun
These objects continuously change direction as they move in a circle. This needs a resultant force to act on the object. This force is the centripetal force. The centripetal force pulls an object toward the centre of the circle.Centripetal force does not exist in its own right, but is provided by the action of other forces. For example, imagine whirling a conker on a piece of string around in a circle. The centripetal force is the result of tension within the string.For a vehicle turning a corner, the centripetal force is provided by friction between the tyres and the tarmac. For objects in orbit, for example the Earth orbiting the Sun, the centripetal force is provided by gravity.
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Acceleration due to centripetal force
An object moving in a circle is constantly changing direction. This means that, even if its speed stays the same, its velocity is constantly changing. (Remember that velocity is speed in a particular direction.)If the object’s velocity is changing, it must be accelerating. The centripetal force is the resultant force that causes this acceleration, and it is always directed towards the centre of the circle.
Without the resultant centripetal force, an object would travel at a constant velocity (constant speed and direction). It would move off in a straight line, as is the case when a hammer-thrower lets go of the hammer.
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Factors affecting centripetal force
The centripetal force needed to keep an object moving in a circle increases if:
the mass of the object increases
the speed of the object increases
the radius of the circle in which it is travelling decreases
Remember: force = mass × acceleration
To maintain a particular circular motion, there will be a particular acceleration. An object with more mass must have more centripetal force acting upon it.
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An object travelling faster covers more distance per second. It will change direction by a bigger angle each second compared to slower object. A greater centripetal force is needed to achieve this bigger acceleration toward the centre.
A circle with a smaller radius has a smaller circumference. Therefore, an object travelling in a circle with a smaller radius has less distance to travel per orbit. It will complete more of the orbit per second, changing direction by a greater angle each second. A greater centripetal force is needed to achieve this bigger acceleration toward the centre.