QUESTION 4CONNECTED PARTICLES NOTES

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Subject: Leaving Certificate Applied MathsTeacher: Nigel MacmillanWeek: Week 11Lesson: Connected Particles Notes

HOMEWORK QUESTION2007 Q3 (b)

A plane is inclined at an angle 45° to the horizontal. A particle is projected up the plane with initial speed u at an angle θ to the horizontal.The plane of projection is vertical and contains the line of greatest slope.The particle is moving horizontally when it strikes the inclined plane. Show that tanθ = 2

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What is Force?

A force is a push or pull upon an object resulting from the object's interaction with another object. Forces only exist as a result of an interaction.

Force is measured in Newtons (N).

Body at Rest

If forces act on a body and it does not move, the forces must balance. Hence, if a number of forces act on a body and it remains at rest, the resultant force in any direction must be zero

Example 1A body is at rest when subjected to the forces shown in the diagram. Find X and Y.

Newton's First Law of Motion states that a body at rest will remain at rest unless an outside force acts on it, and a body in motion at a constant velocity will remain in motion in a straight line unless acted upon by an outside force.

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Example 2

A body moves horizontally at a constant 5ms-1 subject to the forces shown. Find P and S.

Newton’s 2nd Law

The force acting on an object is equal to the mass of that object times its acceleration. This is written in mathematical form as:

Example 3

A body of mass 5kg is acted on by a force of 14N. Find the acceleration

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Example 4

Find the resultant force that would give a body of mass 400g an acceleration of 20ms-1

Gravity and Weight

As we talked about before a body falling under gravity experiences an acceleration of 9.8ms-2. From Newton’s laws it is clear that this acceleration must be carried by a force acting on the body. This force is called the weight of the body

Example 5

Find (a) the weight in newtons of a mass of 23Kg (b) the mass of a box of weight 294N

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Example 6

A bucket of mass 6kg is lowered vertically by a string. Find the force in the string when the bucket is lowered with an acceleration of 3ms-2

Example 8

A piano mass 100kg is lifted up the side of a house. Find the force in the lifting rope when the piano is lifted with an acceleration of 0.25ms-2

Newton’s 3rd Law

For every action, there is an equal and opposite reaction.

This means that for every force there is a reaction force that is equal in size, but opposite in direction

Connected particles

In the following examples we will assume the strings to be light and inextensible. If the surface is said to be smooth there is no resistance to motion i.e. no friction.

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Example 8

A block of mass 3kg is at rest on a smooth horizontal table. The block is connected by a light inextensible string which passes over a smooth pulley. Find a the acceleration of the masses.

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Example 9

Particles of mass 4kg and 2kg are connected by a light string passing over a smooth fixed pulley. The particles hang freely and are released from rest. Find the acceleration of the two particles and the tension in the string.

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Example 10

A body A rests on a smooth horizontal table. The two bodies of mass 2kg and 10kg hanging freely, are attached to A by strings which pass over smooth pulleys. The two strings are taut. When the system is released from rest, it accelerates at 2ms-2. Find the mass of A

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A

2kg 10kg

2ms-1

Pulley systems

In the diagram, pulley A is fixed and pulley B may be raised by pulling down the end X of the string. All the parts of the string not in contact with the pulleys is vertical.For B to move upwards a distance x, a length 2x of string must pass over the pulley A The distance between the the pulley A and the end X of the string is therefore increased by 2xHence if B has an upward acceleration of a, then the end X of the string will have a downward acceleration of 2a

Example 11

Pulley A is a fixed and pulley B has a mass of 4 kg. A load of mass 5kg is attached to the free end of the string. All pulleys are smooth. Find the acceleration when the system is released from rest

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Example 12

A pulley system has loads of 6 kg and 3 kg at the ends of the string, and the moveable pulley has a mass of 2 kg as shown. Find the acceleration of pulley C

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HOMEWORK QUESTION

A light pulley A is suspended from a fixed pulley.Find the acceleration of the 2Kg, 3Kg and 6Kg masses.

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