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# Turning effect of forces

## Moments as turning effect of forces

The moment of a force is the turning effect of a force.

A worker applies a force to a spanner to turn a nut.

A force that is applied to an object away from the object's centre makes the object rotate about a fixed point.

The point of an object that does not change position when it experiences a moment is the pivot.

A moment is a vector quantity because it has a size and a direction. Moments can act clockwise or anti-clockwise.

When you open a door the pivot is the hinge which connects the door to the wall. The force is you pushing the door.

For a seesaw the pivot is the central part which is connected to the ground. The force is the weight of the person sitting on the seesaw.

Both the girl and the boy apply a moment to the seesaw causing it to rotate.

## Moment and torque

A torque $(\vecphy{\tau})$ or moment $(\vecphy{M})$ is the tendency of a force to twist or rotate an object about an axis. The concept of moment can be confused with the momentum $(p=mv).$

The magnitude of $\tau$ is given by the product of the force $F$ and the perpendicular distance $r$ between the point at which the force is applied and the axis of turning: $$\tau=M=rF$$

The SI unit of torque is the Newton metre $(N m)$. It is dimensionally equivalent to the unit of work (i.e. $1\text{ J}=1\text{ N m}$) but describes a different quantity.

When you hold a pen lying on the table at its middle and push it at its tip, the pen will turn around its middle.

$F_1$ on the left acts alone and will change the overall direction of the object as well as creating rotational acceleration. The couple of $F_2$ and $F_3$ creates a moment but no net force.

## Size of moments

The moment of a force can be increased by:

• Increasing the force
• Applying the force further away from the pivot
• Applying the force at right angles to the object

If two people sit at either end of a seesaw and they have different masses, the seesaw will not balance.

If the heavier person sits closer to the pivot than the lighter person, the seesaw could be balanced.

When untightening a nut it is easier to use a longer spanner. This is because you are able to apply a force further from the pivot and so the moment is larger.

Forces $F_{1}$ and $F_{2}$ are equal and are applied at the same point on the beam. Force $F_{1}$ causes a larger moment as it is applied to the beam at right angles.

The beam turns clockwise as $F_{1}$ causes a larger moment than $F_{2}.$

## Calculating moments

The moment of a force can be calculated using the formula $$\Tblue{\text{moment}} = \Tred{\text{force}} \times \Tgreen{\text{perpendicular distance to pivot}}$$

The perpendicular distance is the length of a line drawn between the vector representing a force and the pivot, at right angles to the force.

The perpendicular distance is greatest if the force is at right angles to the object.

The SI unit of a moment is the newton metre and has the symbol $\text{Nm}.$

If you apply a $\Tred{10\text{ N}}$ force on a door at a point $\Tgreen{0.5 \text{ m}}$ away from the pivot, the moment is equal to $\Tred{10\text{ N}} \times \Tgreen{0.5 \text{ m}} = \Tblue{5 \text{ Nm}}.$

The force in the picture is not at right angles to the object. The perpendicular distance of $0.2 \um$ must be used to calculate the moment. Do not use $0.3 \um.$

The force on the beam is on the left hand side of the pivot. It causes a moment of $3 \text{ Nm} \text{ anti-clockwise}.$

## Adding moments and the resultant moment

The resultant moment is the sum of all the different moments on a body.

The resultant moment in the clockwise direction is equal to the sum of the all clockwise moments minus the sum of all the anti-clockwise moments.

The resultant moment in the anti-clockwise direction is equal to the negative of the resultant moment in the clockwise direction direction.

The weight of the load in the wheelbarrow causes a clockwise moment of $\Tblue{100\text{ N}} \times \Tred{0.18 \text{ m}} = \Tblue{18 \text{ Nm}}.$

The force applied by the person holding up the wheelbarrow causes an anti-clockwise moment of $\Tgreen{30\text{ N}} \times \Tred{0.5 \text{ m}} = \Tgreen{15 \text{ Nm}}.$

The resultant moment is $\Tblue{18 \text{ Nm}} - \Tgreen{15 \text{ Nm}} = 3 \text{ Nm}$ clockwise. The person cannot support the load and the wheelbarrow falls to the ground.

## Principle of moments

The principle of moments states that when a body is in equilibrium, the sum of clockwise moments about the pivot is equal to the sum of the anti-clockwise moments about the same pivot.

When the clockwise moments do not equal the anti-clockwise moments there is a resultant moment.

A weight of $160 \text{ N}$ is placed $0.3 \um$ to the left of the pivot. This causes an anti-clockwise moment of $160\text{ N} \times 0.3 \text{ m} = 48 \text{ Nm}.$

A second weight of $240 \text{ N}$ is added to bring the system to equilibrium. The weight needs to be placed so that it causes a clockwise moment of $48 \text{ Nm}.$

The distance between the second weight and the pivot must be equal to $\displaystyle \frac{48\text{ Nm}}{240 \text{ N}} = 0.2 \text{ m}.$

The clockwise moments equal the anti-clockwise moments. The system is in equilibrium.

## Couple and rotational equilibrium

A couple is a pair of equal forces acting on different points on an object so they cause a resulting moment but no net force. That means a couple changes the rotation of an object but it does not change its translational velocity.

When you use a screwdriver, you apply forces with your hand so that the overall location of the screwdriver does not change but it turns on the spot.

When there are forces acting on an object but no net torque, the object is said to be in rotational equilibrium. This means that the clockwise and anticlockwise moments cancel each other out exactly.

$F_1$ on the left acts alone and will cause the object to rotate if pivoted at A. The couple of $F_2$ and $F_3$ causes a rotation around B but does not cause any translation.

## Centre of gravity definition

The centre of gravity of a body is the point where the whole weight of the body seems to act.

The centre of gravity of a regular body is at its centre.

The centre of gravity of an irregular body can be outside the body itself.

If a body is hanging freely, its centre of gravity will always be vertically below the pivot.

A beam is placed on a pivot with a weight at one end. The centre of gravity acts to the right of the pivot and so causes a moment.

If the ruler is in equilibrium, we can use the principal of moments to calculate the weight of the ruler $\Tblue{W}$ \begin{gather*} 0.7 \um \times 100 \text{ N} = 0.5 \um \times \Tblue{W} \\ \Tblue{W} = \dfrac {0.7 \um \times 100 \text{ N}} {0.5 \um} = 140 \text{ N} \end{gather*}

The clockwise moment caused by the centre of gravity cancels out the anti-clockwise moment caused by the weight.

## Centre of gravity and stability

Stability is a measure of a body's ability to return to its original position after it has been tilted slightly.

The stability of vans and lorries is tested so they do not tip over when going round corners.

When a body is tilted, its centre of gravity creates a moment.

If the centre of gravity is high, this moment can cause the body to tilt more and fall over.

If the centre of gravity is low, the moment returns the body back to its original position.

The stability of an object can be improved by lowering the centre of gravity. This can be done by:

• adding weight to the base of the object.
• increasing the area of the base of the object.
Left: The stool is upright and experiences no moment. Centre: The stool is tilted slightly but does not fall. Right: The stool is tilted and falls over.