Embedded Files
Students Learn About:
static friction (with simple calculations)
loads and extension
– load/extension diagram
– tension and compression
stress and strain
– stress/strain diagram
– tension and compression
work, power, energy (without calculations), principle of the conservation of energy
fluid mechanics
– Pascal’s principle
– hydrostatic pressure
– applications to braking systems
Students Learn To:
use mathematical methods to solve simple static friction problems
distinguish between extension, stress and strain
investigate and apply the basic principles of fluid mechanics to simple braking systems
What are the forces of static and kinetic friction?
What are the forces of static and kinetic friction?
Parking your car on the steep hills of San Francisco is scary, and it would be impossible without the force of static friction.
The force of static friction 'Fs' is a force between two surfaces that prevents those surfaces from sliding or slipping across each other. This is the same force that allows you to accelerate forward when you run. Your planted foot can grip the ground and push backward, which causes the ground to push forward on your foot. We call this "grippy" type of friction, where the surfaces are prevented from slipping across each other, a static frictional force. If there were absolutely no friction between your feet and the ground, you would be unable to propel yourself forward by running, and would simply end up jogging in place (similar to trying to run on very slippery ice).
Now, if you park on a hill that is too steep, or if you are being pushed backward by a Sumo wrestler you're probably going to start sliding. Even though the two surfaces are sliding past each other, there can still be a frictional force between the surfaces, but this sliding friction we call a kinetic frictional force. This force of kinetic friction 'Fk', always opposes the sliding motion and tries to reduce the speed at which the surfaces slide across each other. For example, a person sliding into second base during a baseball game is using the force of kinetic friction to slow down. If there were no kinetic friction, the baseball player would just continue sliding (yes, this would make stealing bases in baseball difficult). [Physically, why are there any frictional forces at all?]
Concept Check:
Concept Check:
For each of the following cases of a car changing velocity described in the table below, choose whether it is more likely to be the force of static or kinetic friction causing the change in velocity.
For each of the following cases of a car changing velocity described in the table below, choose whether it is more likely to be the force of static or kinetic friction causing the change in velocity.
What is the formula for the kinetic frictional force 'Fk'?
What is the formula for the kinetic frictional force 'Fk'?
If you press your hands into each other hard and rub them together, the force of kinetic friction will be larger than if you were only pressing your hands together lightly. That's because the amount of kinetic frictional force between two surfaces is larger the harder the surfaces are pressed into each other (i.e. larger normal force 'Fn"
Also, changing the types of surfaces sliding across each other will change the amount of kinetic frictional force. The "roughness" of two surfaces sliding across each other is characterized by a quantity called the coefficient of kinetic friction 'μk'
The parameter 'μk' depends only on the two surfaces in contact and will be a different value for different surfaces (e.g. wood and ice, iron and concrete, etc.). Two surfaces that do not slide easily across each other will have a larger coefficient of kinetic friction 'μk'
We can put these ideas into a mathematical form with the following equation.
What is the formula for the static frictional force 'Fs'?
What is the formula for the static frictional force 'Fs'?
The static frictional force is a little different from the kinetic frictional force. For one, the static frictional force will change its value based on how much force is being applied to the unbudging object. Imagine, for example, trying to slide a heavy crate across a concrete floor. You may push harder and harder on the crate and not move it at all. This means that the static friction responds to what you do. It increases to be equal to and in the opposite direction of your push. But if you finally push hard enough, the crate seems to slip suddenly and starts to move. Once in motion it is easier to keep it in motion than it was to get it started, indicating that the kinetic frictional force is less than the maximum static frictional force.
If you add mass to the crate, say by placing a box on top of it (increasing the amount of normal force 'Fn'), you need to push even harder to get it started and also to keep it moving. Furthermore, if you oiled the concrete (reducing the coefficient of static friction 'μs') you would find it to be easier to get the crate started (as you might expect).
We can put these ideas in a mathematical form by writing the following formula that lets us find the maximum possible static frictional force between two surfaces.
Concept Check:
Concept Check:
What do solved examples involving the force of friction look like?
What do solved examples involving the force of friction look like?
An initially stationary 110kg refrigerator sits on the floor. The coefficient of static friction between the refrigerator and the floor is 0.60, and the coefficient of kinetic friction between the refrigerator and the floor is 0.40. The person pushing on the refrigerator tries to budge the fridge with the following forces.
F push = 400N
F push = 600N
F push = 800 N
For each individual case listed above, determine the magnitude of the frictional force that will exist between the bottom of the refrigerator and the floor.
Solution
Solution
Now that we know the maximum amount of static frictional force is 647N, we know that any force the person exerts below this amount will get matched by the force of static friction.
- loads and extension | stress and strain
Elastic and non elastic materials
Elastic and non elastic materials
Is steel more elastic than rubber? We all have some intuition for elastic and non-elastic materials. But how do we define elasticity? How do we say whether something is more or less elastic? In this video, we will explore the true meaning of elasticity.
Stress & strain
Stress & strain
To measure the elasticity of any material we need to define two quantities. Stress and strain. In this video let's explore what these are and why we define them?
Young's modulus of elasticity
Young's modulus of elasticity
In this video let's explore this thing called 'Young's modulus' which gives a relationship between the stress and strain for a given material.
Terminology
Terminology
strength
strength
Strength: is the ability of a material to withstand applied loads without failure. Strength varies according to the type of load, tensile, compressive, shear or torsional.
ductility
ductility
Ductility: is the ability of a material to be drawn out into thin wire. Not all materials that are ductile are malleable, e.g. lead is very malleable but not very ductile. Another use of the term ductility is a measure of the percentage elongation of a material when undergoing a tension test.
elasticity
elasticity
Elasticity: is the ability of a material to return to its original shape and dimensions after being subjected to a load.
stiffness
stiffness
Stiffness: is the ability of a material to resist elastic deformation under load. This is also measured using Young's modulus (sometimes referred to as the modulus of elasticity).
plasticity
plasticity
Plasticity: is the ability of a material to undergo some degree of permanent deformation without rupture. Processes such as rolling, extruding, pressing, forging, and spinning depend upon plastic deformation and the ability of a material to alter its shape under load. Plasticity usually increases with temperature and this is why many processes are carried out at elevated temperatures. Glass and many other ceramics have no plasticity at room temperature, and therefore behave in a brittle manner.
fatigue
fatigue
Fatigue: is the tendency of a material to break when subjected to repeated cyclic loading where the induced stress is well below the elastic limit.
stress/strain diagram
stress/strain diagram
Load/extension diagrams are a type of graph that displays the relationship between load and extension. Load is plotted on the vertical axis, while extension is on the horizontal axis. The load/extension graph is usually converted into a stress/strain diagram to give a graph that is more easily compared with similar tests for other materials. Both of these graphs will have the same general shape and each display various features that are of importance.
Proportional Limit (From "O" to "A")
Proportional Limit (From "O" to "A")
According to Hooke's Law, Proportional limit (O-A) is the limit where stress is directly proportional to strain. The Stress-strain curve is a straight line (from "O" to "A") within the proportional limit. A material young modulus of elasticity ( ratio of stress and strain) is constant within the proportional limit
Elastic Limit (From "O" to "B")
Elastic Limit (From "O" to "B")
The elastic limit for a material is the limit beyond which the material will not come to its original shape when we remove the external force. The material exhibits elastic properties from point A to B (Yield Point) In the stress-strain curve. If the external load (stress) goes beyond the elastic limit, the material will not come to its original shape.
Upper Yield Point (Point B)
Upper Yield Point (Point B)
Beyond the elastic limit, ductile material exhibits plastic properties. At the upper yield point, the material required the maximum stress to initiate plastic deformation inside the material. The strength of a material corresponding to Point B is known as yield strength.
Lower Yield Point (Point C)
Lower Yield Point (Point C)
After Point C, the material length will increase with a small increase in tensile load (stress). In other words, at Lower Yield Point minimum load is required to exhibit plastic behavior in the material.
Ultimate Tensile Strength (Point D)
Ultimate Tensile Strength (Point D)
A Material has the ultimate tensile strength at Point D on the stress-strain diagram. The ultimate tensile strength of a material is the maximum stress a material can withstand before breaking. After point D, necking starts inside the material
Rapture / Fracture / Breaking Strength (Point E)
Rapture / Fracture / Breaking Strength (Point E)
Point E is the point where material fractures or breaks. Stress at point-E is known as the breaking strength of a material.
Concept Check:
Concept Check:
Explain / Describe each of the features below, in relation to the stress - strain diagram shown (pg 101)
Explain / Describe each of the features below, in relation to the stress - strain diagram shown (pg 101)
Proportional Limit
Elastic Limit
Yield Points
Ultimate tensile strength
Breaking point
Elastic Region
Plastic Region
Uniform Extension
Necking
Work Hardening
Toughness
Resiliance
Stiffness
Stress Formula
Stress Formula
Stress is a measure of the internal reaction that occurs in response to an externally applied load. This internal reaction is related to the original cross sectional area to quantify the nature of the reaction. The unit for stress is the Pascal (Pa) and one Pascal is 1 N/m(2).
Strain Formula
Strain Formula
Strain is the proportional change in length caused when a specimen is under an axial load. Strain is a dimensionless number and may be represented as a ratio or as a percentage.
Concept Check:
Concept Check:
Solve and show your working for the following questions
Solve and show your working for the following questions
A hand brake cable has a tensile load of 4 kN on it. Find the tensile stress is the cable has a cross sectional area of 10mm(2)
A Round d rod in a braking system has a compresive stress of 130 MPa on it. If the load is 350N, what is the rod's diameter
Find the strain of brake rod is it is compressed 0.3mm when it is 70mm long.
If a cable in a braking system has a tensile load placed on it and it has a strain of 0.003, and the length is 1.2m, what is the extension?
- work, power & energy
Work, energy, power & conservation of energy
Work, energy, power & conservation of energy
Work (U) is defined as force (N) multiplied by the distance (d) the force moves. The work done lifting a house brick of mass 3kg from the floor to a bench 1 m high is:
Work (U) = f x d where f = 3 x 10 = 30 N; d = 1 m
= 30 x 1 = 30 J (J = joule which is a unit of work)
If the house brick is held stationery, d = 0 thus no work is done.
Energy has the same units as work because the energy possessed by a body is the work done on the body to give it energy. There are many forms of energy with the most common being mechanical, chemical (TNT, petrol), heat, electrical and atomic energy. In this course we are only concerned with mechanical energy in the following two forms:
Potential energy (PE)
If the brick is held 1 m above the ground the brick has potential energy which is the energy a body has as a result of its position above the earth.
Potential energy = mass (m) × g x height (h) = m x g × h and the unit of energy is the joule (J) which is the same as the unit for work.
The brick held 1 m above the floor is not doing work but it has the potential to do work as a result of its potential energy:
PE = mgh = 3 x 10 x 1 = 30 J
Kinetic energy (KE)
Kinetic energy is the energy a body has because of its motion.
Kinetic energy = 1/2 mass (kg) × velocity² (m/s) If the brick is dropped from the 1 m height the energy of the brick just before hitting the ground is:
KE = 1/2 mv² where m = 3 kg; v² = u² + 2as; v² = 2 x 10 x 1 = 20 m/s
= 0.5 × 3 x 207 = 30 J which is the PE of the brick held at 1m high!
Concept Check:
Concept Check:
Solve and show your working for the following questions
Solve and show your working for the following questions
4 tonne SUV is stationery at the top of a 40° slope that is 20 m long. If it is allowed to roll down the slope, calculate the velocity of the SUV in km/h atthe bottom of the slope, ignoring friction.
Check Copeland Text pg 103. Example 3.7
SOLUTION
SOLUTION
Loss in PE = gain in KE
mgh = 1/2 mv²
V² = 2gh where h = 20 Sin 40° = 12.86 m
V= 16 m/s = 57.7 km/h
- principle of the conservation of energy
Principle of the conservation of energy
Principle of the conservation of energy
This principle states that energy does not disappear when a useful job (ie. work) has been done, it just changes from one form to another or from one object to another object.
In the example above, the KE of the brick just before it hits the ground equals the energy your body provides to lift it to a 1 m height (provided by an input to your body of about 2.5 mg of sugar). When the brick hits the ground it's energy does not disappear - it is transferred from KE to heat energy and strain energy within the brick and floor molecules.
- fluid mechanics
hydrostatic pressure
hydrostatic pressure
If an object is placed into a fluid then the fluid pressure will act upon the entire surface of the object at right angles to the surface. The pressure may be defined as the force per unit area, thus:
The cross sectional size or shape of a container does not affect hydrostatic pressure, but the depth affects it. The deeper something is under a fluid the greater the pressure, for example the deeper a diver is the greater the water pressure on him/her.
Pascal’s principle
Pascal’s principle
We have established that pressure at the bottom of any container is greater than the pressure at the top, and is independent of the shape of a container. It can further be stated that pressure is the same at all points at the same depth. If we increase the pressure at the surface, Po, by forcing a piston into the fluid, then the increase in pressure is the same throughout the liquid. This situation is known as Pascal's principle and is stated as follows:
Pressure applied to an enclosed liquid is transmitted undiminished to every point in the fluid and to the walls of the container.
applications to braking systems
applications to braking systems
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