A-level Mathematics/OCR/M1/Equilibrium of a Particle

Forces

Force is a vector quantity that influences the movement of an object.

Equilibrium of A Particle

If you look at an object lying on a table it is in equilibrium. An object is in equilibrium if the resultant forces acting on it is zero. Yet this does not mean that the object is not moving. It means that the sum of the forces in the x direction is equal to zero: $\sum {f_{x}}=0$ and the sum of the forces in the y-direction is equal to zero: $\sum {f_{y}}=0$.

Example One

A 20 kilogram box is suspended from the ceiling by a rope that weighs 1 kilogram. Find the tension at the top of the rope.

1. The first step is to draw a diagram.

   

2. The second step is to label all the forces. This is the most crucial step. T is the tension force and F is the weight force.

   

3. Now we are ready to solve the problem. Since the object is in equilibrium, because it is not moving, the components must add up to zero. There are no x forces in this problem.

   $\sum {f_{y}}=0=ma-T$

   $T=ma\,$

   Now be careful we need to add the mass of the rope to the mass of the box.

   $T=(20+1)\times 9.8=205.8N$

Friction

It is easiest to calculate the forces on an object that lies on a smooth surface or is not moving. A perfectly smooth surface will not have friction. However in the real world objects are not perfectly smooth, they have rough surfaces. The interaction of these surfaces creates friction. Friction resists motion. Friction is a dimensionless quantity represented by the Greek letter mu $\mu$, $\mu \geq 0$. There are two kinds of frictions; there is statical friction that applies while and object is not moving and there is kinetic friction when an object is moving. Static friction is always greater than kinetic friction. Static friction increases with the force until it reaches a maximum value, this value is known as limiting friction. If there is limiting friction but the object is not moving the object is in limiting equilibrium. With statical friction you need to know these two formulae:

$F\leq \mu R$

$F=\mu R\,$ when the object is in limiting equilibrium

Example Two

A 17 kilogram block is resting on a ramp that makes and angle of 27° with the ground. Calculate how fast the block is accelerating if it is accelerating when a) The surface is smooth b) The surface is rough and $\mu =0.4$.

1. The first step is to draw a diagram.

   

2. Next we need to label all the forces. In this case there will be y and x forces.

a) In the case of a smooth surface, friction has the value of zero.

The weight equals 17 X 9.8 = 166.6 N.

Using F=ma,

17 X 9.8 X cos(90-27)°= 17a

17 X 9.8 X cos 63° = 17 a, so

a=9.8cos63°=4.45 ms⁻² (3sf)

b) The weight equals 17 X 9.8 = 166.6 N

Use F = $\mu$ X R to get F = 0.4 X R