Statics Notes

Centre of gravity is an imaginary point where the entire weight of the body seems to act or may be considered as concentrated.
The centre of gravity is at the same point as the centre of mass for a small body, or the body is in a uniform gravitational field where g is a constant.

For a regular-shaped object, the CG is at its centre of mass (CM).
- Example: the CG and the CM of a metre ruler are located at the mid point of the metre ruler.
Sometimes the CG is found outside the body.
An object with a lower CG is normally more stable than an object with a higher CG from the ground.

Any object at rest may be in one of three states of equilibrium:
- Stable
- Unstable
- Neutral

Using a plumb line with a lamina:
- Hang the lamina freely from a point A and draw a line along the plumb line.
- Hang the lamina freely from another point B and draw the vertical line through B.
- The point of intersection of the two lines (G) is the CG of the lamina.

Centre of mass is the point at which the distribution of mass is equal in all directions and does not depend on gravitational field.
Centre of gravity is the point at which the distribution of weight is equal in all directions and it does depend on gravitational field.

A particle is in equilibrium if the vector sum of the forces on it is zero (all the forces which act upon a particle are balanced).
The forces are considered to be balanced if the rightward forces are balanced by the leftward forces and the upwards forces are balanced by the downward forces.
The condition for a particle to be in equilibrium is that the resultant force acting on the particle is zero.

Static equilibrium – a particle is at rest & not accelerating.
Dynamic equilibrium – a particle is moving & not accelerating.
When a particle is stationary and the resultant force is zero, the particle would continue to be at rest and its velocity v = 0. This is called static equilibrium.
If the particle is moving and the resultant force on it is zero, the particle would continue to move with constant velocity. Its acceleration would be zero. This is called dynamic equilibrium.

When several forces acting on an object are on the same plane, the forces are called coplanar forces.
If the lines of action of all the forces pass through the same point, then the forces are concurrent forces.

The three forces can be represented by a triangle of forces.
The three forces can be arranged either in clockwise or anti-clockwise direction.

A particle acted upon by a number of forces is in equilibrium if the vector sum of the forces is zero.
The principle used in the triangle of forces can be extended for a particle acted upon by more than three forces.
P + Q + R + S + T = 0

If each cable pulls upwards with a force of 25 N, the total upward pull of the sign is 25 \times 2 = 50 N. Therefore, the force of gravity (also known as weight) is 50 N, down.

As the angle with the horizontal increases, the amount of tensional force required to hold the sign at equilibrium decreases.

The product of a force and the perpendicular distance from the line of force to the axis of rotation.
\tau = r \times F (Torque = lever arm x force applied)
\tau (the Greek letter tau) is the symbol for torque.
r is the distance from the pivot point to the point where the force is applied.
F is the magnitude of the force.
\theta is the angle between the force and the vector directed from the point of application to the pivot point.
Torque is a measure of the effectiveness of a force in changing or accelerating a rotation.
The magnitude of the torque is maximum when \theta = 90^\circ. As \sin 90^\circ = 1, therefore \tau = r \times F
Torque is a vector quantity, its direction is perpendicular to both r and F.
The magnitude of torque is \tau = F r \sin \theta (where \theta is the angle between the vectors r and F).
The SI unit of torque is Nm.
- Example: If you push perpendicular to the door with a force of 40 N at a distance of 0.800 m from the hinges, you exert a torque of:
  - \tau = F r \sin \theta = 40 \times 0.800 \times \sin 90^\circ = 32 Nm
  - If you reduce the force to 20 N, the torque is reduced to 16 Nm.
For a couple, torque is calculated as force × perpendicular distance between the lines of action of the forces.
The resultant force acting on the rigid body is zero.
The resultant torque acting on the rigid body is zero.

Statics is the study of forces in equilibrium.
When a particle is in equilibrium, it is either stationary or moving with constant velocity.
If the particle is stationary, then it is in static equilibrium.
If the particle is moving & not accelerating, then it is in dynamic equilibrium.
A particle is said to be in equilibrium if the resultant force acting on the particle is zero.
For a rigid body, two conditions must be met to achieve equilibrium:
- i) the resultant force acting on the rigid body is zero.
- ii) the resultant torque acting on the rigid body is zero.

IMPORTANT FORMULAS!!!!!

Torque:
\tau = r \times F
Where:
- \tau is torque
- r is the distance from the pivot point to the point where the force is applied
- F is the magnitude of the force
- \theta is the angle between the force and the vector directed from the point of application to the pivot point
Magnitude of Torque:
\tau = F r \sin \theta
Where:
- \theta is the angle between the vectors r and F
Resultant Force for Equilibrium:
The sum of forces is zero:
P + Q + R + S + T = 0
Static Equilibrium:
For static equilibrium, the condition is that the resultant force acting on the particle is zero and it is at rest.
Dynamic Equilibrium:
For dynamic equilibrium, the resultant force is zero, and the particle continues moving with constant velocity.