PHSI191 Module 1 - Mechanics (Notes) PDF

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PHSI191 Module 1MechanicsLecture 1KinematicsComponents Of MotionThe components involved in motion involve distance or displacement, speed or velocity, and acceleration.The distance an object travels is defined as the length of the path that the object took in travelling from one place to another (a ...

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PHSI191 Module 1 Mechanics

Lecture 1 Kinematics Components Of Motion The components involved in motion involve distance or displacement, speed or velocity, and acceleration. The distance an object travels is defined as the length of the path that the object took in travelling from one place to another (a scalar quantity). The displacement, however, is the distance travelled, with an associated direction (a vector quantity). The speed of an object is the distance travelled, divided by the time it took to travel that distance (a scalar quantity). The velocity, however, is the displacement vector over the time interval over which the displacement occurs (a vector quantity). Both speed and velocity can be found using the equation:

v=

Δx Δt

Speed and velocity are measured in meters per second (ms-1), distance and displacement are measured in meters (m), and time is measured in seconds (s). Note that sometimes “v” and “Δx” are put in bold to represent that they are the vector quantities, rather than the scalar quantities. Acceleration is a vector which quantified changes in velocity. It is defined to be the rate of change of the velocity:

a=

Δv Δt

Acceleration is measured in meters per second squared (ms-2). Since acceleration is a vector, it has a direction as well as a magnitude. If an object’s acceleration is in the same direction as its velocity, the velocity of the object will increase. If an object’s acceleration is in the opposite direction to its velocity, the velocity of the object will decrease (and possibly reverse direction). If an object’s acceleration is perpendicular to the velocity, the direction of velocity will change, but its magnitude will not (like when an object moves in a circle). For many problems, we are interested in calculating the final velocity of an object when it is undergoing constant acceleration. We can rearrange the equation to do this:

Δv = vf − vi vf = vi + a t

Average Speed Above, we have learned how to calculate the “instantaneous” velocity of an object moving with a constant acceleration (the velocity of an object at a particular instant in time). However, sometimes we will need to use the average speed to solve problems:

vav =

Δx Δt

In this equation, Δx represents the total distance travelled. It can also be calculated for an object undergoing constant acceleration by using:

vav =

1

(vi + vf)

Displacement & Acceleration If an object is undergoing constant acceleration, then the displacement occurring in some given time is:

d = vi t +

1 2 at 2

If an object starts at rest, where the initial velocity is 0, then the equation would simply be:

d=

1 2 at 2

We can also use this equation which does not involve time, which can be useful in solving certain problems:

vf2 = vi2 + 2a d

Acceleration Due To Gravity All objects falling freely towards the earth have the same acceleration. Thus every object in free fall close to the surface of the earth has its downward speed increased by approximately 10ms-1 regardless of its mass. The value of acceleration due to gravity is, therefore, 10ms-2 (g). The velocity of an object in free fall can be found by:

v = gt The direction of this acceleration vector is always going to be towards the centre of the earth.

Vertical & Horizontal Motion Vertical and horizontal motion are independent. For example, what if a cricket ball was thrown at an angle to the vertical? This would mean there is a horizontal velocity and a vertical velocity. The acceleration due to gravity only acts in the vertical direction, and therefore will only change the vertical component of the velocity. It starts at 30ms-1, decelerates after 3s to 0ms-1, changes direction, and then reaches 30ms-1 just as it hits the ground. The ball starts with a horizontal velocity of 10ms-1, and this velocity does not change until the ball ends its flight. This is because there is no acceleration in the horizontal direction, so there can be no change in horizontal velocity.

Lecture 2 Dynamics Newton’s First Law A force causes a change in the motion of an object. For example, if an object is at rest, then we must apply a force to it, to cause it to accelerate and develop a non-zero velocity. This is the basis of Newton’s first law, which states: Any object continues at rest or constant velocity (constant speed in a straight line) unless an external force acts upon it.

The image illustrates the effect of a force on the motion of an object. This object is travelling in a straight line until an external force acts on it in a direction perpendicular to its motion. This causes the object to be deflected from its straight-line motion.

Newton’s Second Law Newton’s second law states: An external force gives an object an acceleration. The acceleration produced is proportional to the force applied, and the constant of proportionality is the mass. The law can be summarised with the equation:

F = ma Force is measured in newtons (N). Since any object of mass near the surface of the earth falls with acceleration due to gravity downwards, it must be acted on by a force:

F = mg This is often referred to as the weight force. It is important to realise that forces are also vectors, so they have a magnitude and a direction. To find the net force on an object, we must find the vector sum of all the individual forces on that object. It is this net force that produces an acceleration. The image shows that the force (F) is equivalent to the vector sum of two other forces (Fx and Fy). Trigonometry can be used to find these forces.

Newton’s Third Law Newton’s third law states: For every action, there is an equal and opposite reaction. This essentially tells us that forces come in pairs. For every force that is applied to an object, there is a force applied by that object. Suppose that a force is exerted by object 1 on object 2. According to this law, object 2 will exert a force of equal size, but in opposite direction, on object 1. This second force is the reaction to the force exerted by object 1. Forces act in pairs and each force acts between a pair of objects. These force pairs are called action-reaction pairs. It is important to correctly identify the action-reaction pairs in a problem and to realise that each force in an action-reaction pair acts on a different object. It is also important to realise that the law does not stop objects from moving. In the diagram, the arrows of the same colour are action-reaction pairs. While Fmatchbox does press back on the finger with a force of equal magnitude and opposite direction to Ffinger, it’s no match for Fmuscles. At the matchbox, the forward force from the finger overcomes the friction force from the table. Each object has an imbalance of forces giving rice to acceleration leftwards.

Circular Motion If an object travels around a circle once, the distance travelled is the circumference of that circle. The time taken is known as

the period (T) of motion. The speed of the object is, therefore:

Δx t 2π r v= T

v=

Since this is the speed of the object, the direction of motion is not important. If the object is traveling at a constant speed, the magnitude of the velocity is constant. The direction that the object is moving in, however, is always changing. Therefore, the velocity of an object in circular motion is continuously changing, since the direction of motion is continuously changing. The velocity of the object will always be tangential to the circle. Since the velocity is changing, the object must be continuously accelerating. This “centripetal” acceleration, is given by:

ac =

v2 r

Since there is a centripetal acceleration associated with the circular motion of an object, there must be a centripetal force to produce it. This force is directed toward the centre of the circular path, along the radius (the same direction as the acceleration). The equation for this is:

Fc =

m v2 r

Sources of centripetal force in everyday life include: - When a car drives in a circle on a flat stretch of road, the centripetal force is supplied by the friction between the rubber tyres and the road surface. - When a ball or some other object is swung in a circle from a string, the tension in the string provides the necessary centripetal force. - A satellite in a circular orbit around the earth is maintained in this circular path by the earth’s gravitational attraction, so gravity provides the centripetal force.

Lecture 3 Types Of Force Centre Of Mass The centre of mass or centre of gravity is a concept used to help solve a lot of physics problems. A large object consists of many parts, and gravity acts downwards on all these parts. But we don’t have to treat all of these parts separately. Every object has a point through which its entire weight appears to act, called the centre of mass.

Friction Friction is the force between two surfaces, parallel to the surfaces. Friction always opposes relative motion. The maximum friction force between two surfaces is given by:

fm a x = μ N The lowercase f indicates friction force, N stands for the normal reaction (usually mg), μ is the coefficient of friction, often less than 1. Friction does not depend on the area, only on N and the nature of the surfaces. Atmospheric friction (or air resistance) is experienced by objects moving rapidly through the air, usually free falling. They

experience a drag (or backwards force) due to having to push the air away. It is because of this that terminal velocity occurs (when the object stops accelerating). Once the atmospheric friction is equal to mg, the net force is 0, so the object will go no faster and continue at the same speed.

Principle Of Moments A torque (τ) may tend to turn a system in the clockwise or anticlockwise direction, depending on the applied force. Torque is defined by:

τ = Fd Torque is measured in newtons per meter (Nm). For a system in static equilibrium, all torques are balanced so that there is no net torque. Therefore the system will remain motionless. This condition is called the principle of moments. The sum of the clockwise moments is equal to the sum of the anticlockwise moments:

τc = τac F1d1 = F2 d 2 Classes Of Lever Levers are divided into 3 classes based on the position of the fulcrum. The fulcrum is the pivot point: - Class 1 - the fulcrum is between the load and the effort. - Class 2 - the fulcrum is at one end of the lever, closest to the load. - Class 3 - the fulcrum is at one end of the lever, closest to the effort.

Lecture 4 Work & Energy Energy Energy is a more abstract concept than the concepts we have talked about so far (such as mass, displacement, velocity, acceleration). Any object which has a certain mass and velocity is described as having a particular kinetic energy. Similarly, an object which has a certain mass and is located at a certain point in a conservative force field (like gravitational or electrical) is described as having a potential energy. The most important thing to remember about energy is that it is always conserved.

Work Work has a specific meaning in physics, and it involves force and displacement. When a force acts on an object, and the object moves a distance in the direction of the force, the work done on the body is:

W = Fd Work is measured in joules (J). Let’s say a man applies a force of 300N to a car, and it moves 10m. Regardless of whether there is friction, or whether the car accelerates, the work done by the man is:

W = 300 × 10 = 3000 The concept of work also applies when force and displacement are not in the same direction. In this case, work is calculated using the component of force in the direction of displacement. The work here is therefore

calculated with the equation:

W = Fcos θ × d

Kinetic Energy The kinetic energy of an object is a measure of the work it can do because of its motion. Consider a single force acting on an object. We define the increase in kinetic energy of the body to be equal to the work done by the force on the body.

The resultant equation we derive from this is:

EK =

1 2 mv 2

Potential Energy In many cases, an external force does work on an object, but that object does not end up in motion. Lifting a box onto a shelf does work on the box. A force is exerted on the box, and the box is displaced in the direction of the force. However, after the force is applied, the box is stationary, sitting on a shelf. The work has been stored in the box as potential energy. We can recover this energy if we allow the box to fall. So essentially, potential energy is the energy an object has because of its position. A clear example of this is gravitational potential energy. Take the example in the image. Work is required to raise the ram against the earth’s gravity. The ram now has the ability to do work (it has potential energy). When the ram is released, the potential energy is converted into kinetic energy, then to work, and then to heat. The equation we use to describe this energy is:

EP = mgh

Conservative Forces Conservative forces result in the conservation of mechanical energy. Non-conservative forces are dissipative, for example, friction. Work done by friction (fd) causes mechanical energy to dissipate, but this energy is not lost. It reappears as an equivalent amount of heat energy (molecules shaking).

Power The definition of work does not refer in any way to the time taken for the work to be done. The same amount of work is done by a runner who sprints up a hill and by a pedestrian who walks up the hill. Clearly, there is an important difference. The rate at which work is done is also an important quantity. This is called the power:

P=

W Δt

Power is measured in watts (W). There is an alternative equation we can use which is helpful if an object is moving at constant velocity while a force is being applied to it:

P = Fv

Efficiency Not all the energy provided to a machine is effectively utilised as work. In real-world cases, some of the energy input is wasted by the machine, due to dissipative forces. There will always be some waste heat or sound generated by a machine. To describe this we can define the mechanical efficiency of a machine as:

η=

work ou t put e n erg y u se d

Lecture 5 Momentum Linear Momentum Linear momentum is a vector quantity, which is its mass multiplied by its velocity:

p = mv Momentum is measured in kilogram meters per second (kgms-1). Newton’s second law can be reformulated to include the concept of momentum:

F=

Δp Δt

This is an indication of how important the idea of momentum is. It tells us that the rate of change of momentum is equal to the net external force. In other words, the greater the force, the greater the change of momentum per unit of time. Here we will define a new quantity, called impulse. Impulse is the name given to the change in momentum:

Δp = FΔt This tells us that the longer the interaction time, the greater the impulse (so the greater the change in momentum).

Collisions There are two broad types of collision: - Inelastic - where momentum is conserved, and kinetic energy is not conserved (some goes to heat, sound). - Elastic - where both momentum and kinetic energy are conserved. Inelastic Collisions A collision is often called totally inelastic if the objects stick together after they have collided. We must define the system to include both objects and nothing else so that there are no external forces. Here, the linear momentum is conserved. But since it is inelastic, their kinetic energy is not conserved. Elastic Collisions Inelastic collisions, we know that both momentum and kinetic energy are conserved. So if we combine both equations, and do some maths, we end up with the equation:

v1i − v2i = − (v1 f − v2 f )

This tells us that the relative speed of the objects before the collision equals the negative of their relative speed after the collision.

Lecture 6 Waves & Oscillations Hooke’s Law For springs and many materials, the length change is proportional to the force (the restoring force):

F = − kx The “k” is the spring constant, measured in newtons per meter (Nm-1) and the negative sign indicates that the force is always in the opposite direction to displacement. As the spring is stretched or compressed by a force, work is done on the spring. Therefore, there must be potential energy stored in the spring:

EP =

1 2

k x2

Simple Harmonic Motion A mass attached to a spring will always experience a restoring force back towards its equilibrium position if displaced. Hence it will oscillate. It will oscillate sinusoidally in simple harmonic motion. The time for each full cycle of oscillation is called the period (T). The frequency is the number of cycles per second:

f =

1 T

Frequency is measured in hertz (Hz) and period is measured in seconds (s). Note that in simple harmonic motion, energy is conserved. When the object is at the maximum amplitude of oscillation, there is no kinetic energy and all the energy is potential. But when the object is at equilibrium, it is at its maximum velocity, so it has all kinetic energy and no potential. At any point in time:

Etotal =

1 2 1 k x + mv 2 2 2

For simple harmonic motion of an object on a spring, the period is:

T = 2π

m k

Another system that undergoes simple harmonic motion is the simple pendulum. The period for a simple pendulum is:

T = 2π

L g

Waves In general, for a wave, the velocity is:

v = fλ Two types of waves exist: - Transverse waves - where the oscillation is perpendicular to the propagation direction (light waves). - Longitudinal waves - where the oscillation is in the direction of propagation (sound waves). Waves can undergo superposition. This is when two waves of the same type meet and overlap, creating a new oscillation. They can undergo constructive interference (left image) or destructive interference (right image), or a mixture of both....