Aslevels Physics Notes
Topic 5: Force Density And Pressure
Types of forces
- The term field in Physics refers to a region of space within which a force is experienced.
- A gravitational field strength g (in N kg-1
) at a point is gravitational
force per unit mass placed at that point. The magnitude of the
gravitational force Fg (in N) on a mass (in kg) at the point is given by:
Fg =mg
The direction of gravitational forces is in the direction of the gravitational field. - An electric field due to a charge is a region of space within which an
electric force is experienced by another charge.
The electric field strength 𝐸 (𝑁𝐶 −¹ ) at a point is the electric force per unit positive charge placed at that point. The magnitude of the electric force on a charge Q is given by: 𝐹𝐸 = 𝑄E
The direction of electric forces on positive charges is in the direction of the electric field. On negative charges, the direction of electric forces is in the opposite direction to the electric field.
Drag forces
- Retarding force as a function of the object velocity is known as drag force. If an object moves faster and faster, the drag force becomes larger and larger
- Aerodynamic and hydrodynamic forces acting opposite to the direction of motion of a solid object relative to earth as cars aircrafts and boats are examples of drag forces.
- In case of viscous drag fluid in a pipe is an examples of drag force. Drag force on the immovable pipe decreases fluid velocity relative to the pipe.
Difference between drag force and friction
- Friction is a rubbing force between two solid objects whereas drag force is between a solid and a fluid.
- Friction is relatively constant at different velocities while drag increases with speed.
- If you rub your hands at different speeds force of friction remains the same although the heat generated will vary, in contrast, as you increase the speed of your bike, the drag increases significantly.
- Friction can be reduced by making the contact surfaces smooth, whereas drag force is decreased on aero plane by smooth surface and streamlined design.
- Drag force is zero when the speed of an object is zero but friction is not zero when an object is at rest
Frictional and viscous forces
- When two solid surfaces are in contact, the forces between them are usually represented as two components:
- Normal contact forces that are perpendicular or normal to the surfaces and
- Frictional forces that are parallel to the surfaces.
- Frictional forces resist the relative motion between the two solid surfaces in contact, i.e. they are always opposite in direction to motion of or the tendency to move.
- Consider a body resting on a horizontal surface with a horizontal force
applied to it, if the magnitude of the force is gradually increased from
zero, there comes a point at which the body just begins to move.
Up to this point, the frictional force on the body just balanced the
applied force and the body does not move, and is usually called static
friction.
The maximum force of friction has been found to be proportional to te magnitude of the normal force.
After the body begins to move, the frictional force is usually called kinetic friction. The magnitude of the kinetic friction is usually smaller than maximum magnitude of the corresponding maximum static friction. - As with all forces, frictional and normal forces occur in pairs and care must be taken to specify which force is being considered.
- A surface is usually described as smooth if there are negligible or no frictional forces between the surface and the body resting on it. The only force between the body and smooth surface would then be the normal contact force.
- Viscous forces in a fluid resist the relative motion of a body through the fluid, i.e. they are always opposite in direction to the motion of the body.
- Consider a body immersed in a fluid with horizontal force applied to it.
Unlike frictional forces, viscous forces are zero when the body’s
velocity is zero, and the body will accelerate upon the application of a
force. Viscous forces increase with the relative velocity of the body in
the fluid, i.e. as the velocity of the body relative to the fluid increases,
the viscous force on the body will increase.
Turning effects of forces
- The ability of a force to rotate a body about a given pivot point is usually called its moment.
- In the algebraic form moment 𝑀 could be expressed as:
- A couple is a pair of forces, equal in magnitude but opposite in directions, and so tends to produce rotation only.
In the algebraic form, torque 𝜏 could be expressed as;
- The torque of a couple is independent of the position of the axis of rotation. Any system of forces acting on a body can usually be reduced to
- A single force acting through a point which only affects translational motion, plus
- A torque of a couple which only affect its rotational motion, although the force and the torque may separately be zero.
- Note the following:
- The term moment usually refers to the turning effect of a net or resultant force while the term torque usually refers to a turning effect without transnational effect. However, these terms are often used interchangeably, and there is no harm in doing so as long as the correct effects are being considered.
- Torque and moment are directed quantities although its direction is limited to clockwise or counter-clockwise in the plane perpendicular to the axis of rotation. It is often regarded as a pseudo vector, with a direction along the axis of rotation determined by the right-hand grip rule – when the fingers of the right hand are wrapped in the direction of the torque, the thumb points in the direction of the torque vector. In some situations, its directions referred to simply as clockwise or counter-clockwise. In other situations, counterclockwise is regarded as positive and clockwise negative.
Equilibrium of forces
- A body is in equilibrium if there is
No resultant force and nor resultant torque acting on it
Conversely, if the resultant force and resultant torque acting on
a body are both zero, the body must be in equilibrium.
The principle of moments states that for any body in equilibrium, the sum
of the clockwise moments about any pivot must equal the sum of the
anticlockwise moments about the pivot.
Conditions for equilibrium
Condition 1 For a body to be in equilibrium, the force vectors must form a
closed polygon. In other words, the resultant force must be zero. Σ𝐹 = 0
Condition 2 For a body to be in equilibrium, the principle of moments applied about any point must be satisfied. In other words, the resultant torque about any point must be zero. Σ𝜏 = 0- If the forces are not parallel, they are usually resolved into (convenient) rectangular components, e.g. x and y components in the x-y plane. The equilibrium conditions are then applied to each component.
- For a body in equilibrium, the choice of pivot point or axis of rotation for calculating net moment is completely arbitrary. The nature of the situation will often suggest a convenient location.
- If only three non-parallel co-planar forces maintain a body in equilibrium, then their lines of action must be concurrent, i.e. all must pass through one point. This can be deduced by considering the point of intersection of any two of the forces. If the third force does not pass through this point, the net moment cannot be zero and the body cannot be in equilibrium. e.g. take the rod acted upon by three co-planar forces as in the figure.
For any point object acted upon by several forces, it is in equilibrium only if the force vectors form a closed polygon.
Resolving forces is another way of checking if
forces on a point object balance out. Any force
may be resolved into two perpendicular
components. Essentially, this means that the
force vector may be replaced by two components at right angles to each
other. Suppose one of the components is to be parallel to a line at angle 𝜃 to the line of action of the
force, as in figure. The
other component is
therefore at right angles
to the first line. The
parallel component is
𝐹𝑐𝑜𝑠𝜃, and the
perpendicular component is 𝐹𝑠𝑖𝑛𝜃. 
Now consider a point object acted upon by several forces. Resolve each force vector into two perpendicular components along and at right angles to a given line. This is shown in the figure where the given line is x-axis, so each force is resolved into x and y-component.
If the forces balance out, the
1. The x-components balance out
−𝐹1𝑐𝑜𝑠𝜃₁ + 0 + 𝐹2𝑐𝑜𝑠𝜃₂ = 0
𝑠𝑜 𝐹₁𝑐𝑜𝑠𝜃₁ = 𝐹2𝑐𝑜𝑠𝜃₂
2. The y-components balance out
𝐹₁𝑠𝑖𝑛𝜃₁ − 𝐹₃ + 𝐹₂𝑠𝑖𝑛𝜃₂ = 0
𝑠𝑜 𝐹₃ = 𝐹₁𝑠𝑖𝑛𝜃₁ + 𝐹₂𝑠𝑖𝑛𝜃₂
Centre of gravity
- While a body is not a point mass, but is composed of many point masses (atoms and molecules), the vector sum of the weights of all the point masses can usually be taken as acting at a single point.
Density and pressure
- Pressure is defined as the normal force per unit area. 𝑝 = 𝐹 𝐴 Its SI unit is pascal (Pa) or newton per square meter. It is a scalar quantity. Other commonly used units include the atmosphere (atm), the millimeter of mercury (mm Hg) and the bar. 1 atm (standard) =760 mm Hg=1.01 bar = 1.01x10⁵Pa
- Density 𝝆 is defined as mass per unit volume, i.e. 𝜌 = 𝑚 𝑉 . Its SI unit is kgm-3 . It is a scalar quantity.
- A fluid is a substance that can flow; it follows that both liquids and gases are fluids. The pressure p at depth h in a fluid of density 𝜌 is given by the equation: 𝑝 = 𝜌𝑔ℎ . the equation could be derived from the definition s of pressure and density as follows;
It should be noted that the pressure p is independent of the shape of
the area and the shape of any container holding the fluid.
- The equation gives the pressure p due to the fluid alone. If the surface of the liquid is subjected to pressure 𝑝𝑠 , from another fluid, e.g. the surface of water subjected to air pressure, then the pressure 𝑝𝑠 , should be added,i.e. 𝑝 = 𝜌𝑔ℎ + 𝑝𝑠
- The difference in pressure Δ𝑝 within the fluid between two points
separated by a vertical height Δℎ is given by:
It follows that all points at the same depth Δℎ = 0 in a fluid are at the same pressure.
- Upthrust is the upward (or buoyant) force exerted by a fluid on a
body when the body is completely or partially immersed in the fluid.
According to Archimedes’ principle,
When a body is immersed in a fluid it experiences an Upthrust equal in magnitude to the weight of fluid displaced.
i.e. Magnitude of Upthrust = weight of fluid displaced
Upthrust acts through what was the centre of gravity of the fluid before the fluid was displaced. The direction of Upthrust is always upwards, i.e. opposite to the direction of free fall. Upthrust arises because the pressure in a fluid increases with depth and thus pressure on the lower surface is greater than that on the upper surface. It could be explained by considering a body immersed in a fluid. E.g. the cylinder shown in the given diagram, submerged in a fluid. The upward force (𝐹1)
- It is essential to note that the Upthrust is independent of the shape of the body and the shape of any container holding the fluid. This can be shown by considering the equilibrium of a bag of the fluid with a similar shape and volume as a body to be immersed. With the bag of fluid in equilibrium. The Upthrust U on it must be equal in magnitude and opposite in direction to its weight 𝑤𝑓. When the bag of fluid is replaced by the body to be immersed, the body will experience the same Upthrust U.
- If a body floats partially immersed in a fluid, then
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