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Moments (Mechanics)

Moments

The turning effect of the force on the body on which it is acting is measured by the moment of a force.

The moment of a force depends on the magnitude of the force and the distance from the axis of rotation.

The moment of a force about a point is (the magnitude of the force) × (the perpendicular distance of the line of action of the force from the point).

When there are several forces acting on a body the moments about a point can be added so long as a positive direction (clockwise or anticlockwise) is specified and is considered for each moment.

Worked Example: Finding the moment

Find the moment of F about P

Find the moment of F about P in the above diagram. Find the moment of F about P when θ=35,F=8 and d=14.

Solution

Moment of F about P is F×dsinθ clockwise. Here, we have found the perpendicular distance from P using trigonometry and multiplied it by the magnitude of the force. When describing a moment you need to give a direction of rotation. Therefore when we are given the values this is Moment=F×dsinθ,=8×14sin(35),=64.241Nm (to 3d.p.). The moment of the force is measured in newton-metres Nm, therefore the moment of F about P is 64.241Nm.

Sum of moments

The diagram shows a set of forces acting on a light rod. Calculate the sum of the moments about the point P.

Solution

Each force is already perpendicular to the point P.

The moment of the 6N force is 6×2=12Nm anticlockwise.

The moment of the 14N force is 14×2=28Nm clockwise.

The moment of the 5N force is 5×(2+3)=25Nm anticlockwise.

Total clockwise =28Nm and total anticlockwise =37Nm. Therefore the sum of the moments is 3728=9Nm anticlockwise. As the anticlockwise total was greater we chose anticlockwise as the positive direction.

Worked Example: Finding the moment when the distance given is not perpendicular

Find the moment of the force about P

Find the moment of the force about P in the above diagram.

Solution

The perpendicular distance from P to the force will be the distance opposite the angle given by 7sin(60). Therefore The moment of the force=11×7sin(60),=77sin(60),=66.684Nm anticlockwise (3 d.p.).

Worked Example: Bodies resting in equilibrium

Finding the normal reactions

Suppose that there is a uniform rod of length 9m and weight 30N. It rests on a support X at one end point and a support Y which is 5m away from support X. Calculate the magnitude of the reaction at each of the supports.

Solution

We can draw a diagram showing all the forces acting.

The weight of the rod will act at its centre of mass - as it is a uniform rod the weight acts at the mid-point. The rod is in equilibrium so the total forces acting upwards will equal the total forces acting downwards when we resolve vertically. RX+RY=30. Consider the moments about point X, we have that clockwise moments will equal anticlockwise moments therefore 30×4.5=RY×(4.5+0.5),135=5RY,27N=RY. Now we can use this value to find RX RX+RY=30,RX=3027,=3N. Therefore the reaction at point X is 3N and the reaction at point Y is 27N.

Finding the centre of mass in a non-uniform body

Suppose that there is a non-uniform rod AB of length 10m and weight 15N. It is in a horizontal position and rests on supports at points C and D. The distance AC=3m and AD=7m. The magnitude of the reaction at C is four times the magnitude of the reaction at D. Find the distance of the centre of mass of the rod from A.

Solution

We can draw a diagram showing all the forces acting.

Here we have assumed that the centre of mass acts at a point x m from point A. As the rod is not uniform we can not say that the centre of mass is at the mid point of the rod. We resolve vertically 4R+R=15,5R=15,R=3. Now we take moments about point A 15×x=(4R×3)+(R×7),=36+21,=57,x=3.8. The centre of mass is 3.8m from A.