Topic 2 - Mechanics

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[edit] Topic 2: Mechanics

===2.1.1=== Define displacement, velocity, speed, and acceleration.

Displacement: A vector quantity that describes a position of a particle in reference to an origin. It has both magnitude and direction.

Velocity: The total speed of a particle in a given direction.

Speed: The rate of change in position without regard to direction.

Acceleration: The rate of change of velocity.

===2.1.2=== Define and explain the difference between instantaneous and average values of speed, velocity, and acceleration.

An instantaneous value is a value taken at an instant.

An average value is taken over a period of time.

===2.1.3=== Describe an object’s motion from more than one frame of reference.

If one car, travelling at 30 ms-1, overtakes another travelling at 25 ms-1¬, then according to the driver of the slower car, the relative velocity of the faster car is 5 ms-1. This is moving from one frame of reference to another. The velocities of 30 ms-1¬ and 25 ms-1¬ were from the position of a stationary observer. We moved from this frame of reference into that of one of the drivers.

===2.1.4=== Draw and analyse distance-time graphs, displacement-time graphs, velocity-time graphs and acceleration-time graphs.

Displacement-time graphs:

• Gradient = velocity
• Area under the graph = nothing significant

Velocity-time graphs:

• Gradient = acceleration
• Area under the graph = displacement

Acceleration-time graphs:

• Gradient = Change in Acceleration
• Area under the graph = not really significant (the amount of acceleration, total velocity changed)

===2.1.5=== Analyse and calculate the slopes of displacement-time graphs and velocity-time graphs, and the areas under velocity-time graphs and acceleration-time graphs. Relate these to the relevant kinematic quantity. See previous pages.

===2.1.6=== Determine the velocity and acceleration from simple timing situations.

Velocity = Displacement/Time

Acceleration = ΔVelocity/Time

===2.1.7=== Derive the equations for uniformly accelerated motion. Let

   t be time for the body accelerates
   a be acceleration
   u be the initial speed
   v be the final speed
   s be the distance travelled by object in time t

First Formula v = u+at

As the change of velocity, plus the initial equals to the final velocity.

Second Formula s=ut+¹/₂at²

As the displacement is the area under the velocity-time graph.

Image:V-t.jpg

Third Formula v²=u²+2as

From the first equation we can derived it as t=^v-u/_a. Substituting it to the second equation we get s=u^v-u/_a+¹/₂a(^v-u/_a)²

Expending it we see the third formula.

===2.1.8=== Describe the vertical motion of an object in a uniform gravitational field.

An example of uniformly accelerated motion is the vertical motion of an object in a uniform gravitational field. If we ignore the effects of air resistance, this is known as free-fall. In the absence of air resistance, all falling objects have the same acceleration of free-fall, independent of their mass.

===2.1.9=== Describe the effects of air resistance on falling objects.

As the object's velocity increases, the force of air resistance, which is opposite in direction to the movement of the object, becomes greater and greater until it reaches a value equal to the value of the force of gravity. This eliminates a net force put upon the object, which causes it to fall at a constant velocity, no longer accelerating. This is called terminal velocity.

===2.2.1=== Describe force as the cause of deformation or velocity change.

A force is recognised by the effect it produces, and can cause an object to deform (change shape), speed up, slow down, and change direction.

===2.2.2=== Identify the forces acting on an object and draw free-body diagrams representing the forces acting.

Forces should be labeled with a name or symbol – for example, weight, normal reaction, friction, etc. Vectors should have lengths approximately proportional to their magnitudes. Free-body diagrams are just a simple diagram of one object, for example, a book or a person, and all of the forces acting must make contact with the object somewhere and must be named.

Image:Freebodydiagram.png

===2.2.3=== Resolve forces into components.

A straight vertical or horizontal force do not have components. Diagonal forces, however, do. A 4N force in a NW direction will have both a horizontal and vertical component. An angle should be given, so use this in trigonometry to work out the sides (which are the components).

Example Two forces act on particle P. Find the magnitude of the net force acting in the horizontal direction and the magnitude of the net force acting in the vertical direction and hence find the resultant force acting on P.

===2.2.4=== Determine the resultant force in different situations.

Adding of vectors is usually needed here. To find the resultant vector, join the beginning of the first onto the end of the last. See 1.4.4 for subtracting (although all you need to do is reverse the vector to be subtracted and add it to the other). The resultant vector may need to be worked out through trigonometry but sometimes even just Pythagoras can be used.

===2.2.5=== Describe the behaviour of a linear spring and solve related problems.

When a force is applied to a spring (through the addition of a mass or simply pulling it) a tension force is produced. The spring increases in length. The difference between the natural length and stretched length is called the extension of a spring.

As you pull the spring, the further you extend it, the greater the force you have to exert in order to extend it even further.

Hooke’s Law, after Robert Hooke, states that up to the elastic limit [region of proportionality] the extension of a spring is equal to the tension force, F. The constant of proportionality k is called the spring constant. The SI units for the spring constant [k] are Nm-1. F = kx

Up to the elastic limit, the force is proportional to extension, but beyond this point proportionality is lost. If this point is passed, the spring can become permanently deformed in such a way that when weights are removed the spring cannot go back to its original length.

===2.2.6=== State Newton’s first law of motion.

An object continues in a state of rest or uniform motion in a straight line unless acted upon by an external force.

===2.2.7=== Describe examples of Newton’s first law.

Examples of Newton's First Law:

• A parachutist in free-fall (if the force of weight is larger than air friction the parachutist accelerates downwards; as they get faster, air friction increases until weight = friction. Then the parachutist is at constant velocity – the acceleration is zero – resultant force is zero)
• A car traveling (if air resistance equals the force forwards due to engine, then the car is at constant velocity – no resulting force. If either force is larger than the other, then there is a resultant force, and the car accelerates)
• A book at rest on a table (acceleration = zero; resultant force = zero; thus it is at rest (no resultant force)
• Lifting a heavy suitcase (if the suitcase is too heavy to lift, it is not moving therefore acceleration = zero. Thus the pull from the person plus the reaction from ground is equal to the weight of the suitcase)

===2.2.8=== State the conditions for translational equilibrium.

If the resultant force on an object is zero then it is said to be in translational equilibrium, or just equilibrium. To be in translational equilibrium, an object must be constantly at rest or moving with uniform velocity in a straight line. There are essentially two types of [translational] equilibrium.

Static equilibrium: Static equilibrium exists when an object is at rest. Take the example of the book on the table.

Dynamic equilibrium: Dynamic equilibrium exists when an object is moving at constant velocity in a straight line. Take the example of the car traveling.

===2.2.9=== Solve problems involving translational equilibrium.

Basically, when an object is in translational equilibrium, it means that the forces that are acting on the object cancel each other out. They are still acting, but are equal to each other – the net force, or resultant vector, is zero. So when a car moves at constant speed, the force pushing it forward by the engine is equal to that of air resistance pushing it back. The weight of the book is equal to the normal reaction from the surface of the table.

=== 2.2.10 === State Newton’s second law of motion.

Acceleration is directly proportional to the force acting and is in the same direction as the applied force.

F = ma

But because sometimes mass of a system doesn’t remain constant (like a firework rocket, sand falling onto a conveyer belt etc) it can be helpful to express the law in a more general form:

F = ^Δ(mv)/_t

Because p = ^mv/_t, F = ^Δp/_Δt

Thus, Force = Change of momentum over Change in time.

===2.2.11=== Solve problems involving Newton’s second law.

Example Problems:

• The diagram shows a block of wood of mass 1kg attached via a pulley to a hanging weight of mass .5 kg. Assuming that there is no friction between the block and the bench and taking g to be 10 ms-2, calculate the acceleration of the system.

The force acting on the system is the weight of the hanging mass which is 0.5g = 5N. Using Newton’s second law F = ma, we have 5 = (1.5)a a = 3.3 Hence, the acceleration is 3.3 ms-2

• A person of mass 70kg is strapped into the front seat of a car, which is travelling at a speed of 30ms-1¬¬. The car brakes and comes to rest after travelling a distance of 180m. Estimate the average force exerted on the person during the braking process.

===2.2.12=== State Newton’s third law of motion.

To every action, there is an equal and opposite reaction

KEY POINTS ABOUT THIS LAW:

• The two forces in the pair act on different objects – this means that equal and opposite forces that act on the same object are NOT Newton’s third law pairs. *Not only are the forces equal and opposite, but they must be of the same type. In other words, if the force that A exerts on B is a gravitational force, then the equal and opposite force exerted by B on A is also a gravitational force.

===2.2.13=== Discuss examples of Newton’s third law.

Examples of Newton's Third Law

• Forces between roller skaters (if one pushes off another, they both feel a force, equal and opposite, but their acceleration will be different due to mass)

A roller skater pushes off from a wall (the force on the girl by the wall causes the girl to accelerate backwards. The mass of the wall (and earth) is so large that the force on it does not cause any acceleration)

• A rocket in space propels gasses at high velocity in one direction, establishing a force. This causes the rocket to move in the opposite direction with a force of equal magnitude that the gasses have.

===2.3.1=== Define inertial mass.

Inertial mass [is the property of an object that] determines how it responds to a given force, whatever the nature of that force. It essentially measures a body’s inertia. Different masses have different accelerations when a force acts on them (take the first roller-skater above).

===2.3.2=== Compare gravitational mass and inertial mass.

Gravitational mass [is the property of an object that] determines how much gravitational force it feels when near another object.

Different masses have different gravitational forces acting between them. The two concepts are different, but gravitational and inertial mass are equivalent – a body’s gravitational mass is equal to its inertial mass. The fact that different objects have the same value for free-fall acceleration shows this.

===2.3.3=== Discuss the concept of weight.

Weight is a force. The ‘weight of an object’ is a force (N). Mass and weight are often confused. Mass is the amount of matter contained in an object, whereas weight is a force acting on the object. However, there is ambiguity as to the definition of weight even to physicists. It is generally defined in two ways:

• (a) the gravitational force on an object, mg
• (b) the reading on a supporting scale (ie, scales to ‘weigh’ yourself)

Although these two definitions are the same if the object is in equilibrium, they are different in non-equilibrium situations.

For example: if both the object and the scale were put into a lift and the lift accelerated upwards then the definitions would give different values. It is safer to use ‘gravitational force’ instead of weight.

Gravitational force = mg

On the surface of the Earth, g is approximately 9.81 ms^-2, so to work out ‘weight’ or ‘gravitational force’ you multiply mass in kg by 9.81 ms^-2.

===2.3.4=== Distinguish between mass and weight.

Mass is generally defined as the amount of matter contained in a body, although this is difficult, because what is matter, and how do we quantify it? Weight is also ambiguous, although it is agreed that it is a force. Mass is measured in kg, for example a 7 kg block, and weight is a force, for example, a weight of 10 N.

If an object were taken to the moon, its mass would be the same, but its weight would be less because the gravitational forces on the moon are less than on Earth. On Earth the two terms are muddled because they are proportional. For example, double the mass and you double the weight. People talk about losing weight when they really want to do is lose mass.

===2.4.1=== Define linear momentum and impulse.

Linear momentum is the product of mass and velocity. P = mv. Momentum is a vector, and the units are kg ms^-1.

Impulse is the change in momentum, in any situation, particularly if it happens quickly. (∆p = F ∆t).

===2.4.2=== State the law of conservation of linear momentum.

The law of conservation of linear momentum states that

The total linear momentum of a system of interacting particles remains constant provided there is no resultant external force.

===2.4.3=== Derive the law of conservation of momentum for an isolated system consisting of two interacting particles.

Image:Elastic Collision.gif

===2.4.4=== Solve problems involving momentum and impulse.

Momentum is a vector quantity. The outcome of a collision depends on the mass of each particle, their initial velocities, and how much energy is lost in the collision. But whatever, the outcome, momentum is always conserved. Any predicted outcome which violates the conservation of linear momentum will not be accepted.

Example Problems

• A railway truck, B, of mass 2000kg is at rest on a horizontal track. Another truck, A, of the same mass moving with a speed of 5 ms-1 collides with the stationary truck and they link up and move off together. Find the speed with which the two trucks move off and also the loss of kinetic energy on the collision.

• Suppose that in the previous example that after the collision truck A and truck B do not link and that after the collision, truck A is moving with a speed of 1ms-1 and in the same direction as prior to the collusion. Find the speed of truck B after the collision as well as the kinetic energy lost on collision.

[edit] Elastic and inelastic collisions.

Inelastic collisions are when mechanical energy is lost when two objects collide but momentum is conserved.

Elastic Collisions happen when there is no mechanical energy lost or momentum lost in a collision.

In the real world mechanical energy is always lost during a collision. But some do approximate quite well to being elastic. The collision of two pool balls is almost elastic, as is that between two steel ball bearings.

Example A car of mass 1000 kg is parked on a level road with its handbrake on. Another car of mass 1500kg travelling at 10ms-1 collides with the back of the stationary car. The two cars move together after collision in the same straight line. They travel 25m before finally coming to rest. Find the average frictional force exerted on the cars as they come to rest.

===2.5.1=== Define work.

Work is done when a force moves its point of application in the direction of the force. If the force moves at right angles to the direction of the force, then no work has been done.

Work usually involves a transfer of energy from one form to another (ie kinetic to gravitational). The amount of energy transferred is equal to the work done. Work has been done against a force. Work is equal to the force multiplied by the distance moved – OR work is equal to the magnitude of the force multiplied by the displacement in the direction of the force.

W=FΔs

Work is a scalar in Nm or Joules; work done by a system is positive; work done on a system is negative.

===2.5.2=== Determine the work done by a non-constant force by interpreting a force-displacement graph.

A force-displacement graph in this regard will probably be one about applying a force to a spring. The area under the graph is the work done, which is then ½ Fs. But in this case F = ks, which means that the work done is ½ X (ks) X s = ½ ks2. The work that has been done is stored in the spring as elastic potential energy, Eelas. So, Eelas is ½ ks2. This idea can be extended to find the work done by any non-constant force. If we know how the force depends on displacement then to find the work done by the non-constant force we just calculate the area under the force-displacement graph.

===2.5.3=== Solve problems involving the work done on a body by a force.

Remember to use W = force X distance (this is essentially it)

Example Problems:

A force of 100 N pulls a box of weight 200 N along a smooth horizontal surface as shown below.

Calculate the work done by the force (a) in moving the box a distance of 25 m along the horizontal and (b) against gravity.

===2.5.4=== Define kinetic energy. Kinetic energy is the energy that a body possesses by virtue of its motion. The formula is ½ mv² – the kinetic energy a body tells us how much work the body is capable of doing.

===2.5.5=== Describe the concepts of gravitational potential energy and elastic potential energy.

If an object of mass m is lifted to a certain height h above the surface of the earth then the work done is mgh and the object now has a potential energy equal to the work done mgh (force = mg, displacement = h). For example, if you place an object at rest on top of a wall it has potential to do work, because if it falls off the wall onto a nail sticking out of a piece of wood then it could drive the nail further into the wood. Work is needed to lift the object on to the top of the wall and we can think of this work as being ‘stored’ as potential energy in the object. The object has gravitational potential energy by virtue of its position in the Earth's gravitational field ∆Ep = mg ∆h. No matter where an object is placed in the Universe it will be attracted by the gravitational force of Earth. When an object is moved a distance h in the Earth’s gravitational field and in the direction of the field, its change in gravitational potential energy is mgh (provided that g is constant). Elastic potential energy is similar. Work must be done to stretch a spring or similar object, and this work done is stored in the spring as elastic potential energy – if you let it go or remove the mass, it will spring back, converting this elastic potential energy into kinetic energy. Eelas = ½ ks2.

===2.5.6=== State the principle of conservation of energy.

There are several ways of stating this principle:

• Overall the total energy of any closed system must be constant.
• Energy is neither created nor destroyed, it just changes form.
• There is no change in the total energy of the Universe.

===2.5.7=== List different forms of energy and describe examples of the transformation of energy from one form into another.

• Thermal energy

Can be used to boil water and produce steam. The kinetic energy of the molecules of steam (thermal energy) can be used to rotate magnets and this rotation generates an electric current. The electric current transfers the energy to consumers where it is transformed into, for example, thermal and light energy (filament lamps) and kinetic energy (electric motors).

• Chemical energy

This is associated with the electronic structure of atoms and therefore with the electromagnetic force. An example of its transformation is in combustion in which carbon combines with oxygen to release thermal energy, light energy, and sound energy.

• Nuclear energy

An example of this is the splitting of nuclei of uranium by neutrons to produce energy.

• Electrical Energy

• Solar Energy

Different forms of energy all fall into the category of either potential or kinetic energy and are all associated with one or other of the fundamental forces.

===2.5.8=== Define power.

Power is the rate at which energy is transferred. This is the same as the rate at which work is done. The unit for power is Js^-1 or Watt. 1W = 1 Js^-1.

Power = energy transferred/Time = work done/Time

If something is moving at a constant velocity v against a constant frictional force F, the power P needed is P = Fv

===2.5.9=== Define and apply the concept of efficiency.

Efficiency is the ratio of useful energy to the total energy transferred.

The easiest equation for this is Efficiency = W_u\W_a

W_u = mgh = Useful Work W_a = Fs = Actual Work

Depending on the situation, we can categorize the energy transferred (aka work done) as useful or not. In a light bulb, the useful energy would be light energy; the ‘wasted energy’ would be thermal energy (and non visible forms of radiant energy).

Since it is a ratio it has no units. Often it is expressed as a percentage.

===2.5.10=== Solve work, energy, and power problems.

See following attached pages.

Formulas in formula booklet are:

• W = Fscosθ
• Power = work / time = Fv

Efficiency is not in there.

===2.6.1=== Draw a vector diagram to show that the acceleration of a particle moving with uniform speed in a circle is directed toward the centre of the circle.

Velocity changes from VA to VB; the magnitude of the velocity stays the same, but the direction changes – thus, the particle is experiencing acceleration.

===2.6.2=== State the expression for centripetal acceleration.

The expression for centripetal acceleration is

a_c = ^v2/_r

===2.6.3=== Identify the force producing circular motion in various situations.

We can find the expression for the centripetal force F by using Newton’s second law, F = ma, so that, with

a_c = v²/r

we have:

F = mv²/r

where m is the mass of the particle.

In order for the particle to move in a circle a force must act at right angles to the velocity vector of the particle and the speed of the particle must remain constant. This means that the force must also remain constant.

The effect of the centripetal force is to produce acceleration towards the centre of the circle. The magnitude of the particle’s linear velocity and the magnitude of the force acting on it will determine the circular path that a particular particle describes.

These are some situations in which centripetal forces arise:

• Gravitational force.
• Frictional forces behind the wheels of a vehicle and the ground.
• Magnetic force.
• Tension force in a string whirling an object around your head.

===2.6.4=== Solve problems for particles moving in circles with uniform speed.