# Mechanics - Ms Gugger's Classes 2016 MECHANICS STATIC OF A PARTICLE 6.2 Inertial mass, momentum, including a change of momentum (conservation of momentum and impulse are not required), force, resultant force, weight action and reaction Equations of motion using absolute units (equations of motion should be described from a diagram, showing all the forces acting on the body, and then writing down the equation of motion. Extensions could include cases involving a system of two or move connected particles. Examples are to be restricted to rectilinear motion, including motion on an inclined plane.) Motion of a body, regarded as a particle under the action of concurrent coplanar forces (the case of equilibrium should be regarded as an

application, where net force is zero). DEFINITIONS Statics The study of the equilibrium of a particle under the action of concurrent coplanar forces Particle A body or object with mass such that all the mass acts through one point. Large objects such as cars, trains are regarded as point particles. Always assumed that a particle is uniform or a point particle meaning that in problems a square or rectangle cam be drawn to represent the object

Measurements The MKS system The description of motion is dependent on the measurement of length, mass and time. Length - metre Mass kilogram Time second

You may come across other units convert these into the units above. Note: Mass of an object is the amount of matter it contains. Measurement of the mass of an object does not depend on its position. In maths, the terms mass and weight do not mean the same thing. Particle model Vector and scalar quantities Object is consider as a point. Length, mass and time are scalar quantities specify magnitude

This can be done when the size of the object can be neglected in comparison with other lengths in the problem being considered, or when rotational motion effects can be ignored only. Position, displacement, velocity and acceleration are vector quantities must be specified by both magnitude and direction. Forces Units of force Force: push or pull motion. An action which tries to alter the

One unit of force is the kilogram weight (). If an object on the state of the motion of a particle. Eg. Pushing car with required force will move it. If force is not great enough, the car wont move. Has both magnitude and direction hence a vector quantity. Expressed as and is the magnitude of vector . surface of the Earth has a mass of , then the gravitational force acting on the object is . The standard unit of force is the newton (). The conversion is = , where is the acceleration due to gravity.

The significance of this unit will be discuss in the upcoming section. Resultant Force Since force is a vector quantity, the vector sum of the forces acting at a point is called the resultant force DEFINITIONS - CONTINUED Equilibrium A system of concurrent coplanar forces acting on a particle is a set of two-dimensional forces all passing though the same point. The equilibrium of a particle means that the vector sum of all the forces acting on the particle is zero. The forces cancel each other out; the particle is said to be

in static equilibrium, and the acceleration of the particle is zero. Lamis Theorem States that if three concurrent forces act on a body in equilibrium the magnitude of each force is proportional to the sine of the angle between the other two forces When a particle is in equilibrium and is acted upon by only two forces, and : This implies that the two forces act in a straight line but in opposite directions.

Redraw the diagram with the force forming a closed triangle. The magnitude of each force is proportional to the sine of the angle between the other two force. If a particle is in equilibrium and has two forces acting on it that are not in a straight line, then there must be another/third force that balances these forces. When a particle is acted upon by three forces, , and , and the particle is in equilibrium, the vector sum of these three forces must be zero: In this case these three concurrent forces can be represented as the sides of a closed triangle. Vector sum when the forces are placed head to tail is zero. The sine rules gives:

Trigonometry and Pythagoras Theorem can be used to solve many force problems In this section answers should be given as exact values where appropriate; otherwise they should be given correct to decimal places. and therefore Lamis Theorem is: EXAMPLE - CAMBRIDGE Example 1 Example 2

Find the magnitude and direction of the a) Four forces are acting on a particle as shown. Express the resultant force in form. b) Give the magnitude of the resultant force and the angle that it makes with the -direction. resultant force of the forces and acting on a particle at as shown in this diagram. EXAMPLES - JACPLUS

Worked Example 1 Worked Example 2 particle is in equilibrium and is A acted upon by three forces. One force acts horizontally and has a magnitude of ; another force acts at right angles to the first force and has a magnitude of . A particle is in equilibrium and is acted upon by three forces. The first force acts horizontally and has a magnitude of ; the second force, of magnitude , acts at with the first force.

Find the third force. Find the third force. NOTE: if forces are not at right angles, then the sine and cosine rules can be used to solve the force problem OTHER TYPES OFF FORCES Weight force Mass vs Weight Mass is a scalar quantity. Scenario 1: object placed on bathroom scales

the reading is the amount of mass in the object. Scenario 2: object suspended from a spring balance, reading gives a measure of the gravitational force exerted the objects weight. Weight is a vector quantity, defined by a downward force exerted in a gravitational field on a particle. Particle of mass m kg, this weight force is or and always acts vertically downwards, where is the acceleration due to gravity in the Earths gravitational field. A weight force acts on all bodies near the Earths surface. The units for a weight force can be either kilograms-weights or newtons; the SI unit for force is the newton.

Resolution of a force in a given direction Scenario Model railway trolley Tension is the same at all points in the string and this tension is unaltered if the string passes over a smooth hook or pulley. Note: only light string is consider at this stage therefore mass of the string can be ignored. String is assumed inextensible therefore the string is taut. When , the trolley moves along the track. As increases, the trolley still moves along the track, but the same

force will have a decreasing effect on its motion, ie the acceleration of the trolley will be less. An alternative method to Lamis Theorem and using trigonometry is called the method of resolving forces. Consider a force making an angle of with the positive -axis. Resolving a vector meaning splitting the vector up into its horizontal and vertical components:

is the unit vector in the direction s the unit vector in the direction When , the trolley stays in equilibrium, ie If at rest it will not move, unless the force is strong enough to cause it to topple sideways. A force acting on a body has an influence in directions other than its line of action, except the direction perpendicular to its line of action. Let the force of N be represented by the vector .

Tension - denoted by T Set on smooth tracks, pulled by a force of magnitude N along a horizontal string which makes an angle of with the direction of the track. Resolving forces Let be the resolute of in the direction and let be the perpendicular resolute. From the triangle of vectors, it can be seen that

As the force represented by does not influence the movement of the trolley along the track, the net effect of on the movement of the trolley in the direction of the track is . The force represented by is the resolved part of the force in the direction of . Resolution of a Force The resolved part of a force of N in a direction which makes an angle with its own line of action is a force of magnitude . NOTE: the resolved part is also called the component of the force in the given direction. Vector sum of components - In two dimension when a particle is at equilibrium: component: Resolved horizontal components of the forces is

equal to zero direction component: Resolved vertical components of the forces in a perpendicular direction is equal to zero direction Resolving all the forces Lamis Theorem can only be used when dealing with problems involving three forces. When there are more than three forces, acting, resolve the forces.

EXAMPLE JACPLUS A HOMEWORK TASK COMPARISON BETWEEN METHODS Example 3 Solve using trigonometry Example 6 solve using method of resolution of vectors particle of mass is supported by A two strings of lengths and . The ends of the string are attached to two fixed points apart on the same horizontal level. A particle of mass is supported by

Find the tensions in the strings. Find the tensions in the strings. two strings of lengths and . The ends of the string are attached to two fixed points apart on the same horizontal level. EXAMPLE - JACPLUS Worked Example 7 string is tied to two fixed points, and , on A the same horizontal plane. A mass of is suspended from the string at by means of a smooth hook that is pulled aside by a horizontal force of N until the system is in

equilibrium. The parts of the string and are then inclined at angles of and to the vertical respectively. Find the value of and the tension in the strings. EXAMPLES - CAMBRIDGE Example 3 Example 4 Find the resolved part of each of the Find the component of the force in the direction of following forces in the direction of

the vector . a) Example 5 Forces of , and act at a point as shown in the diagram. b) Find: a) The magnitude of the resultant of these forces b) The direction of the resultant force with respect to the force.

INCLINED PLANES AND CONNECTED PARTICLES 6.3 Inertial mass, momentum, including a change of momentum (conservation of momentum and impulse are not required), force, resultant force, weight action and reaction Equations of motion using absolute units (equations of motion should be described from a diagram, showing all the forces acting on the body, and then writing down the equation of motion. Extensions could include cases involving a system of two or move connected particles. Examples are to be restricted to rectilinear motion, including motion on an inclined plane.) Motion of a body, regarded as a particle under the action of concurrent

coplanar forces (the case of equilibrium should be regarded as an application, where net force is zero). INTRODUCTION If all forces under consideration are acting in the same plane, then these forces and the resultant force can each be expressed as a sum of its and components. If a force acts at an angle of to the axis, then can be written as the sum of two forces, Worked Example 13 Cambridge A particle at is acted on by forces of

magnitude and as in Example 1 from Cambridge. If the particle has mass , find the acceleration and state the direction of the acceleration. One horizontal One vertical Worked Example 14 Cambridge The force is resolved into two components: The component is parallel to the axis The component is parallel to the axis

A block of mass is pulled along a horizontal plan by a force of inclined at to the plane. The coefficient of friction between the block and the plane is Find the acceleration of the block. FORCES AND INCLINED PLANES Action and Reaction forces Newtons Third Law of Motion states that for every action there is and equal and opposite reaction. That is, if a body exerts a force by resting on a desk, then the desk exerts a force of equal magnitude by in the opposite direction.

This is called the Normal Reaction denoted by . The Normal Reaction is always perpendicular to the surface and only acts when a body is in contact with another surface. Normal reaction forces for inclined planes For a mass on a plane that is inclined to the horizontal, the normal reaction force is at right angles to the plane In such a situation, it is often advantageous

to choose the direction up the plane to be and the direction perpendicular from the plane to be . Note: The incline plane is smooth indicating that there is no frictional forces acting. Two external acting forces which are perpendicular to each other: Normal Reaction The weight force, which acts vertically downward. EXAMPLES Worked Example 15 Cambridge particle of mass lies on a smooth

A plane inclined at to the horizontal. There is a force of acting up the plane. Find the acceleration of the particle down the incline and the reaction force . Worked Example 10 - Jacplus A mass of lies at rest on a smooth plane inclined at to the horizontal. The mass is held in place by a string that is parallel to the plane and passes over a smooth pulley at the top of the plane. The other end of the string is connected to a mas

of that hangs vertically Find the value of . EXAMPLE HOMEWORK TASK Worked Example 11 - Jacplus a rope fastened to a horizontal is beam at and . A mass of is suspended from and a mass of is suspended from . The section of the rope makes an angle of with , and section is inclined at below the horizontal, while section is inclined at an angle of to the vertical. Find the tensions in all the sections

of the rope and the value of . DYNAMICS 6.4 Inertial mass, momentum, including a change of momentum (conservation of momentum and impulse are not required), force, resultant force, weight action and reaction Equations of motion using absolute units (equations of motion should be described from a diagram, showing all the forces acting on the body, and then writing down the equation of motion. Extensions could include cases involving a system of two or move connected particles. Examples are to be restricted to rectilinear motion, including motion on an inclined plane.) Motion of a body, regarded as a particle under the action of concurrent coplanar forces (the case of equilibrium should be regarded as an

application, where net force is zero). DYNAMICS THE STUDY OF THE CAUSES OF MOTION OF PARTICLES Weigh t The gravitational force per unit mass due to the Earth is newtons per kilogram. It varies from place to place on the Earths surface, having a value of at the poles and at the equator. We use to represent . A mass of on the Earths surface has a force of acting on it.

This force is known as the weight. Momentum Concept is fundamental to Newtons Second Law of Motion. Momentum of a body of mass moving with velocity is defined by Standard unit for momentum is NOTE: momentum is a vector quantity Worked Example 6 - Cambridge a) Find the momentum of a particle of mass moving with Worked Example 12 - Jacplus Find the magnitude of the momentum of a car of mass moving at .

velocity b) Find the momentum of a particle moving with a velocity of in an easterly direction. Worked Example 7 - Cambridge Find the change in momentum of a ball of mass if the velocity changes from to . The ball is moving in the one direction in a straight line. DEFINITIONS Newtons laws of motion Jacplus Newtons Laws of Motion Cambridge Newtons First Law of Motion:

Newtons First Law of Motion: Every body continues in a state of rest or A particle remains stationary, or in uniform uniform motion in a straight line unless acted upon by an external force Newtons Second Law of Motion: The resultant force acting on a body is proportional to the rate of change of momentum. Newtons Third Law of Motion: For every action there is an equal and

opposite reaction. straight-line motion (i.e in a straight line with a constant velocity), unless acted on by some overall external force, i.e. if the result force is zero Newtons Second Law of Motion: A particle acted on by forces whose resultant is not zero will move in such a way that the rate of change of its momentum w.r.t time will at any instant be proportional to the resultant force Newtons Third Law of Motion: If a particle exerts a force on a second

particle , then exerts a collinear force of equal magnitude an opposite direction on . IMPLICATIONS OF NEWTONS LAWS OF MOTION CAMBRIDGE The First Law A force is needed to start an object moving (or to stop it), but once moving the object will continue at a constant velocity without any force being needed The Second Law Let represent the resultant force exerted on an object of mass moving at a velocity in a straight line.

Then: If an object is at rest or in uniform straight- line motion, then any forces acting on the object must balance meaning the resultant force is zero If motion is changing (in speed or direction), then the forces cannot balance meaning the resultant force is non-zero Assuming that the mass is a constant: The newton is the unit of force chosen that the constant is equal to when the mass is

measured in kilograms and the acceleration in . That is, one newton is the force which causes a change of momentum of . Newtons Second Law of Motion When measuring force in Newtons , mass in kilograms and acceleration in , the formula can be written as: Note: The directions of the acceleration and the resultant force are the same. WORKED EXAMPLES CAMBRIDGE Example 8 Example 9 Example 10

A stone of mass is acted on by a ice-hockey puck of mass An loses speed from to over a distance of . Find the uniform force which causes this change in velocity. Three forces , and act on a particle of mass , where and . force of . What will be its acceleration? How much further could the puck travel?

The acceleration of the particle is . Find . WORKED EXAMPLES JACPLUS Worked Example 13 particle of mass on a smooth horizontal A plane is acted upon by three forces in that horizontal plane. has a magnitude of and acts in direction , has a magnitude of and acts in direction , and has a magnitude of and acts in . If is a unit vector of magnitude acting each and is a unit vector of magnitude acting north, find the magnitude of the acceleration of the particle.

IMPLICATIONS OF NEWTONS LAWS OF MOTION CAMBRIDGE The Third Law An alternative wording Newtons third law is: If one object exerts a force on another (action force), then the second object exerts a force (reaction force) equal in magnitude but opposite in direction to the first. It is important to note that the action and reaction forces, which always occur in pairs, act on different objects. If they were to act on the same object, then there would never be accelerated motion, because the resultant force on every object would be zero.

For example: If a person kicks a door, then the door acceleration open because of the force exerted by the person. At the same time, the door exerts a force on the foot of the person which decelerates the foot. For a particle hanging from a string, the forces and both act on the particle. They are not necessarily equal and opposite forces. In fact, they are equal only if the acceleration of the particle is zero (by Newtons Second Law). The forces and are not an action-reaction pair of Newtons third law, as they both act

on the one particle. If a person is pulling horizontally on a rope with a force , then the rope exerts a force of on the person Normal reaction force If a particle lies on a surface and exerts a force on the surface, then the surface exerts a force, , on the particle. If the surface is smooth, this force is taken to act at right angles to the surface and is called the normal reaction force. In such a situation we have

Sliding Friction By experiment, it has been shown that the magnitude of the frictional force, , on a particle moving on a surface is given by: Where is the magnitude of the normal reaction force and is the coefficient of friction. The frictional force acts in the opposite direction to the Surfaces velocity of the particle. Coefficient of Friction If the particle is on a platform which is being accelerated upwards at then If a particle of mass lies on a smooth surface and a

force of F N acts at an angle of to the horizontal, then . Worked Example 11 Cambridge A box is on the floor of a lift that is accelerating upwards at . The mass of the box is Find the reaction of the floor of the lift on the box . Rubber tyre on dry road Approaches 1 Two wooden surfaces 0.3 to 0.5

Two metal surfaces 0.1 to 0.2 If the surface is taken to be smooth, then . Worked Example 12 Cambridge A body of mass at rest on a rough horizontal plane is pushed by a horizontal force of for seconds. a) If , how far does the body travel in this time? b) How much further will it move after the force is removed? EXAMPLES JACPLUS Worked Example 14

Worked Example 16 A boy of mass on a skateboard of mass A boy of mass rides a bicycle up a slope is freewheeling on a level track. His speed decreases from to as he moves a distance of . Find the magnitude of the resistant force, which is assumed to be constant. of in . If the resistance to the motion is one-quarter of the weight force, find the tractive force of the tyres when he

ascends with an acceleration of . DYNAMIC WITH CONNECTED PARTICLES 6.5 Inertial mass, momentum, including a change of momentum (conservation of momentum and impulse are not required), force, resultant force, weight action and reaction Equations of motion using absolute units (equations of motion should be described from a diagram, showing all the forces acting on the body, and then writing down the equation of motion. Extensions could include cases involving a system of two or move connected particles. Examples are to be restricted to rectilinear motion, including motion on an inclined plane.) Motion of a body, regarded as a particle under the action of concurrent

coplanar forces (the case of equilibrium should be regarded as an application, where net force is zero). CONNECTED PARTICLES WITH EXAMPLES Consider the following: Light rope being pulled from each end. Has mass. Newtons laws of motion state: . The tension in the string is equal magnitude in both sections. The inclined plane is rough.

The body on the inclined plane is accelerating up the plane. At every point on rope: There are two forces: equal and opposite Both have magnitude of . Two particles connected by a taut rope moving on smooth plane The body is accelerating down the inclined plane A smooth light pulley (ie. The weight of the pulley is considered

negligible and the friction between the rope and pulley is negligible). The tensions in both sections of the rope can be assumed to be equal. Two masses on a smooth inclined plane. In general,. EXAMPLES - CAMBRIDGE Worked Example 17 Worked Example 18 car of mass tows another car

A of mass with a light tow rope. If the towing car exerts a tractive force of magnitude newtons and the resistance to motion can be neglected, find the acceleration of the two cars and the tension in the rope. The diagram shows three masses of kg, kg and kg connected by light inextensible strings, one of which passes over a smooth fixed pulley. The system is released from rest. Calculate: a)

The acceleration of the masses b) The tension in the string joining the 4kg mass to the kg mass. c) The tension in the string joining the kg mass to the kg mass. EXAMPLE - JACPLUS Worked Example 18 Two boxes of masses 5kg and 10kg

are connected by a string and lie on smooth horizontal surface. A force of 30 newtons act on the 10kg box at an angle of to the surface. Find the acceleration of the system and the tension in the string connecting the boxes. VARIABLE FORCES 13E (CAMBRIDGE) Inertial mass, momentum, including a change of momentum (conservation of momentum and impulse are not required), force, resultant force, weight action and reaction Equations of motion using absolute units (equations of motion should be

described from a diagram, showing all the forces acting on the body, and then writing down the equation of motion. Extensions could include cases involving a system of two or move connected particles. Examples are to be restricted to rectilinear motion, including motion on an inclined plane.) Motion of a body, regarded as a particle under the action of concurrent coplanar forces (the case of equilibrium should be regarded as an application, where net force is zero). Reminder: Where , and are the position, velocity and acceleration at time respectively. EXAMPLES - CAMBRIDGE Worked Example 19 Worked Example 20

A body of mass , initially at rest, is particle of mass units moves in a A straight line and, at time , its position relative to a fixed origin and its speed is acted on by a force of newtons, where (seconds). Find the speed of the body after seconds and the distance travelled. a) If the resultant force is , and and when , find in terms of .

b) If the resultant force , and when find when . EQUILIBRIUM 13F Inertial mass, momentum, including a change of momentum (conservation of momentum and impulse are not required), force, resultant force, weight action and reaction Equations of motion using absolute units (equations of motion should be described from a diagram, showing all the forces acting on the body, and then writing down the equation of motion. Extensions could include cases involving a system of two or move connected particles. Examples are to be restricted to rectilinear motion, including motion on an inclined plane.)

Motion of a body, regarded as a particle under the action of concurrent coplanar forces (the case of equilibrium should be regarded as an application, where net force is zero). EQUILIBRIUM RECAP If the resultant force acting on a particle is zero, the particle is said to be in equilibrium. The particle has zero acceleration: If the particle is at rest, it will remain at rest. If the particle is moving, it will continue to move with constant Lamis Theorem - revised Lamis Theorem is a trigonometric identity which simplifies problems involving three forces acting on a particle in equilibrium when the angles between the forces are known. Let N, N and N be forces acting on a particle,

forming angles with each other as shown. velocity. Triangle of forces Forces can be represented by vectors If the particle is in equilibrium, then: and arranged to form a triangle as seen to the right, if they (, and ) act on a particle in equilibrium. Particle in equilibrium, implies: Proof: Represent the forces as a triangle as shown Magnitudes of forces and angles

between forces can be found using: Sine Rule now gives: trigonometric rations (if the triangle contains a right angle) Ie Sine or cosine rule. Resolution of forces For three forces If and only if and For coplanar forces, we can show that the resultant force is by showing that the sum of the resolved parts in each of two perpendicular direction is .

Worked Example 23 EXAMPLES - CAMBRIDGE Worked Example 21 Find and in the system of forces in equilibrium as shown in the diagram. Forces of magnitude N, N and N act on a particle in equilibrium. a) Sketch the triangle of forces to represent the three forces. b) Find the angle between the N and N forces, correct to decimal places

Worked Example 24 Three forces of N act on a particle as shown in the diagram. Show the particle is in equilibrium by resolving in the and directions. Worked Example 22 Forces of magnitude N, N and N act on a particle which is in equilibrium, as shown in the diagram. Find the magnitude of . Worked Example 25 The angles between three forces of magnitude N, N and N acting on a particle are and respectively. Find and , given that

the system is in equilibrium. VECTOR FUNCTIONS - 13G Inertial mass, momentum, including a change of momentum (conservation of momentum and impulse are not required), force, resultant force, weight action and reaction Equations of motion using absolute units (equations of motion should be described from a diagram, showing all the forces acting on the body, and then writing down the equation of motion. Extensions could include cases involving a system of two or more connected particles. Examples are to be restricted to rectilinear motion, including motion on an inclined plane.) Motion of a body, regarded as a particle under the action of concurrent coplanar forces (the case of equilibrium should be regarded as an

application, where net force is zero). EXAMPLES - CAMBRIDGE Worked Example 26 Worked Example 27 Forces and act on a particle of mass At time , the position of particle of mass is Find: Find: which is at rest. given by .

a) The acceleration of the particle a) The initial position of the particle b) The position of the particle at time , given that initially it is at the point b) The Cartesian equation describing the path of the particle

c) The Cartesian of the path of the particle. c) The resultant force acting on the particle at time Assume system of units.