Density
The density of a substance is defined as its mass per unit volume:
ρ = m / v
Unit: kg m-13
Okay so it doesn't explicitly state in the spec that we need to know methods of determining density but I have a feeling we should so here goes. We need to measure the mass and volume of the object. Mass can be measured using a digital balance. We can calculate the volume of regular shapes (e.g a cuboid) by taking measurement with a ruler or we can submerge irregular shapes in a measuring cylinder of water and see how much the water is displaced. Now do ρ = m / v.
Pressure
Pressure is the normal contact force exerted per unit cross-sectional area:
p = F / A
Unit: N m-2
Archimedes' principle
We need to know about pressure in liquids and gases too, not just in solids. Gases and liquids exert pressure on surfaces because of the constant bombardment by their molecules.
We can calculate the pressure exerted by a vertical column of a liquid using:
p = hρg
The buoyant force on submerged objects can be explained in terms of pressure differences at its upper and lower surfaces.
Force at the top surface = hρgA
Force at the bottom surface = (h + x) ρgA
Resultant upward force = (h + x)ρgA - hρgA
Upthrust = xρgA
The volume of the object is Ax and it has mass Axρ so the upthrust is equal to the weight of the fluid displaced. This is Archimedes' principle, 'when an object is immersed fully or partially in a fluid, the upward force applied by the fluid on the object is the same as the weight of the fluid displaced by the object'.
If upthrust is less than weight the object will sink. For floating objects upthrust must equal the weight of the object.
The moment of a force is the turning effect of a force about an axis or point. It equates to the product of force and the perpendicular distance of the line of action of the force from the axis or point of rotation:
moment = F x
Units: N m
When a body is in equilibrium the net force acting on it is zero and its net moment is zero. The principle of moments can be used to solve problems where an object is in rotational equilibrium. The principle of moments states that for a body in rotational equilibrium the sum of the anticlockwise moments abut any point is equal to the sum of the clockwise moments about the same point.
A pair of equal but opposite forces that are parallel and act on different lines is known as a couple. The moment of a couple is known as a torque. The torque of a couple is the product of one of the forces and the perpendicular separation between the forces:
torque of a couple = Fd
The centre of mass of an object is a point through which any externally applied force produces straight line motion with no rotation. The centre of gravity of an object is an imaginary point were the entire weight of an object appears to act. A freely suspended object will come to rest with its centre of gravity vertically below the point of suspension. We can use a plumb-line to experiment about the centre of gravity (for 2-D objects).
When an object is in equilibrium there is no resultant force acting on it. We can demonstrate the forces in a free-body diagram using a triangle of forces:
- arrows are drawn to represent each of the three forces end to end (ie the end of one arrow is the beginning of the next)
- the triangle is closed if the object is in equilibrium. this is because the et force is zero and so the object is in equilibrium.
If an object is moving through a fluid it will experience a drag force from the fluid. Drag is a frictional force that opposes the motion of the object. Its magnitude is mainly affected by:
- the speed of the object
- the cross-sectional area of the object
- the shape of the object
- the density of the fluid
- the texture of the object
Objects with larger cross-sectional areas will experience a greater drag force. This is because drag ∝ speed2.
Often, we call the drag force experienced by objects in air air resistance. We are able to reduce air resistance. For example, modern vehicles have streamlined shapes as this increases their top speed and reduces their fuel consumption.
During a vertical fall (e.g through air) the weight of an object will of course remain constant but the speed will increase so the drag force will increase. At the instant objects start to fall the object will experience no drag force and the total force is equal to the weight (as this is the only force acting on the object). As the object falls its speed increases so drag increases. The resultant force on the object decreases (as drag opposes the motion ie is in the opposite direction to weight). At terminal velocity, drag = weight and the object has zero acceleration and its speed is constant.
We can investigate the motion of an object falling affected by drag force by using a motion sensor and data-logger:
- attach the object to a light polystyrene ball using a thin thread passing over a pulley.
- Drop the object through a cylinder of liquid (e.g water or glycerol). This will pull the polystyrene ball vertically upwards (the motion of this ball is equal to the motion of the object)
- when the ball is moving up with constant velocity, the object is moving down with terminal velocity
The mass of an object depends on the amount of matter it contains. The net/resultant force acting on an object will make it accelerate in that particular direction. The net force (F):
F = m a
Force is measured in newtons (N). A force of 1 newton will give a 1kg mass an acceleration of 1m s-1 in the direction of the force.
The weight of an object on the surface of the Earth is the gravitational force acting on the object:
W = m g
The weight of an object can be calculated using a newtonmeter. These are calibrated to show the gravitational force acting on an object in newtons (e.g an object of mass 1kg will show a weight of 9.8N).
The easiest way to analyse the forces acting on an object is to draw free-body diagrams (these isolate all the forces acting on a particular object. Each force vector is represented by an arrow labelled with the force it represents, each arrow is drawn to the same scale (ie the longer the arrow the greater the force).
Tension, normal contact force, upthrust, and friction are all examples of forces that can be added to a free-body diagram.
