Forces

Key points

1.5.1 Effects of forces

• A force is a push or a pull. Force is measured in newtons (N).
• A force can change the size of an object. It can also change its shape. Forces can stretch, squash, bend or twist an object.
• The load on a spring is the force that stretches it. Often the load is the weight of masses hung on the spring.
• The extension of a spring is how much longer it gets: extension = stretched length − original length.
• Measuring extension (the experiment):
  1. Hang the spring from a clamp stand. Fix a metre rule next to it.
  2. Measure the original length of the spring with no load.
  3. Hang one mass on the spring. Read the new length on the rule. Keep your eye level with the bottom of the spring (or its pointer), to avoid a parallax error.
  4. Work out the extension. Find the load from the mass using W = mg (see 1.3).
  5. Add more masses, one at a time, and repeat.
  6. Remove the masses. Check that the spring goes back to its original length.
• Plot a load–extension graph. For a spring, the first part of the graph is a straight line through the origin. This means the extension is directly proportional to the load: double the load, double the extension.
• For larger loads the graph curves. The extension is no longer proportional to the load.
• Extra detail: the rule “extension is proportional to load” is often called Hooke’s law.
• The resultant force is the single force that has the same effect as all the forces acting on the object together.
• For forces along the same straight line: add the forces that act in the same direction. Subtract the forces that act in opposite directions. Always give the direction of the resultant.
• Example: the driving force on a ferry is 5000 N forwards. Water resistance on the ferry is 3000 N backwards. The resultant force is 5000 − 3000 = 2000 N forwards.
• If the forces on an object are balanced, the resultant force is zero.
• If there is no resultant force, an object at rest stays at rest. An object that is moving keeps moving in a straight line at constant speed.
• A resultant force changes the velocity of an object. It can change the object’s speed (faster or slower). It can change the object’s direction of motion. It can change both.
• Solid friction is the force between two surfaces that are touching. It can oppose (impede) motion: it acts against the way the surfaces slide, or try to slide, over each other.
• Friction produces heating. When the surfaces rub, work is done against friction. This transfers energy to the internal (thermal) store of the surfaces, so they get warmer (see 1.7).
• Drag is friction on an object moving through a liquid (a fluid). Example: water resistance on a swimmer or a ferry.
• Drag also acts on an object moving through a gas. Drag in air is called air resistance.
• The spring constant k is the force per unit extension: k = F / x. Its unit is N/m or N/cm. Extended
• A stiffer spring has a larger spring constant. It needs a larger force for each metre (or centimetre) of extension. Extended
• The limit of proportionality is the point on a load–extension graph where the straight line ends and the graph starts to curve. Extended
• Up to the limit of proportionality, the extension is proportional to the load. Beyond it, the extension is not proportional to the load. Extended
• k = F / x applies only up to the limit of proportionality. Extended
• F = ma: the resultant force equals mass × acceleration. Extended
• The force and the acceleration are always in the same direction. If the resultant force is opposite to the motion, the acceleration is opposite to the motion too, so the object slows down. Extended
• Extra detail: one newton is the resultant force that gives a mass of 1 kg an acceleration of 1 m/s².
• Motion in a circle: an object moves in a circle when a force acts on it perpendicular (at right angles) to its motion. This force points towards the centre of the circle. Extended
• A force perpendicular to the motion changes the direction of motion but not the speed. The direction keeps changing, so the velocity keeps changing. Extended
• Examples: on a roundabout, friction from the road on the tyres pulls a motorbike towards the centre. For a ball whirled on a string, the tension in the string pulls the ball towards the centre. Extended
• For an object moving in a circle: Extended
  • If the force increases and the mass and radius stay the same, the speed increases.
  • If the force increases and the mass and speed stay the same, the radius decreases.
  • If the mass increases, a larger force is needed to keep the same speed and radius.
• If the force towards the centre is removed and no other resultant force acts, the object moves off in a straight line at constant speed, along the tangent to the circle. Extended
• A real ball on a string that breaks starts to move along the tangent. Then its weight pulls it downwards, so its path curves. Extended

1.5.2 Turning effect of forces

• The moment of a force is a measure of its turning effect about a pivot. A pivot is the point that an object turns about.
• Everyday examples of moments:
  • Opening a door: the handle is far from the hinges (the pivot), so a small force has a large turning effect.
  • A spanner: a longer spanner puts the force further from the nut, so the same force gives a larger moment. This is why a long spanner loosens a tight nut more easily.
  • A seesaw: a lighter child can balance a heavier child by sitting further from the pivot.
  • A wheelbarrow, scissors and a bottle opener also use moments.
• moment = force × perpendicular distance from the pivot. The unit of moment is the newton metre (N m).
• The perpendicular distance is the shortest distance from the pivot to the line of action of the force. The line of action is the line along which the force acts. Use the distance measured at right angles to the force.
• A moment is either clockwise or anticlockwise.
• Principle of moments: when an object is balanced, the total clockwise moment about a pivot equals the total anticlockwise moment about the same pivot.
• Example: a beam is balanced on a pivot. One force acts on each side. Then force on the left × its distance = force on the right × its distance.
• An object is in equilibrium when there is no resultant force and no resultant moment on it. Both conditions are needed.
• With more than one force on each side of the pivot: add up all the clockwise moments. Add up all the anticlockwise moments. In equilibrium, the two totals are equal. Extended
• The weight of a uniform beam acts at its centre. If the pivot is not at the centre, the beam’s own weight has a moment about the pivot. Include it. Extended
• In equilibrium there is also no resultant force. So the upward force from the pivot equals the total of the downward forces. Extended
• Experiment to show there is no resultant moment in equilibrium: Extended
  1. Balance a metre rule on a pivot at its centre, with no loads on it. Its weight then acts at the pivot and has no moment about it.
  2. Hang masses at different distances on both sides of the pivot. Hang them exactly on the marks. Use more than one mass on at least one side.
  3. Move one mass until the rule balances again.
  4. Record each load (W = mg) and its distance from the pivot.
  5. Calculate the total clockwise moment and the total anticlockwise moment.
  6. The two totals are equal (within the limits of experimental accuracy). So there is no resultant moment.
  7. Repeat with different loads and distances.

1.5.3 Centre of gravity

• The centre of gravity of an object is the point where its whole weight seems to act.
• For a uniform, regular object (such as a metre rule or a square card) the centre of gravity is at its centre.
• Finding the centre of gravity of a thin, flat, irregular shape (a plane lamina):
  1. Make three small holes near the edge of the shape, spaced well apart.
  2. Hang the shape from a pin through one hole, so that it can swing freely.
  3. Hang a plumb line (a string with a small weight) from the same pin.
  4. When the shape stops swinging, mark the line of the string on the shape.
  5. Repeat from a second hole.
  6. The centre of gravity is where the two lines cross.
  7. Hang the shape from the third hole to check: the third line should pass through the same point.
• Why this works: a freely hanging object comes to rest with its centre of gravity directly below the pin. Then its weight has no moment about the pin, so it does not turn.
• The stability of an object is how hard it is to knock over.
• An object is more stable when its centre of gravity is low and its base is wide.
• Why: when an object is tilted, its weight acts along a vertical line through the centre of gravity.
  • If this line is still inside the base, the weight has a moment that turns the object back onto its base.
  • If this line falls outside the base, the weight has a moment that turns the object further over. The object topples.
  • With a low centre of gravity and a wide base, the object can be tilted further before the line falls outside the base.
• Examples: a motorbike on its side stand is stable while the line of its weight falls between the tyres and the stand. A bus with heavy luggage on its roof has a higher centre of gravity, so it topples more easily on a slope. A desk lamp has a heavy, wide base so that it does not fall over.

Common mistakes

• Students use the stretched length of the spring as if it were the extension. / The mark scheme wants the extension: stretched length − original length.
• Students give the unit of moment as N/m. / The mark scheme wants N m (newton metre): force multiplied by distance.
• Students use a distance that is not measured from the pivot, or not at right angles to the force. / The mark scheme wants the perpendicular distance from the pivot to the line of the force.
• Students describe equilibrium as “the forces are balanced” only. / The mark scheme wants both: no resultant force and no resultant moment.
• Students put one force into F = ma when several forces act. / The mark scheme wants the resultant force: add or subtract all the forces along the line first. Extended
• Students write that an object moving in a circle is pushed outwards. / The mark scheme wants a force towards the centre, perpendicular to the motion, which changes the direction of motion. Extended

Exam tips

• State what a resultant force can do: change the speed, the direction of motion, or both.
• Describe the spring experiment as numbered steps: original length, add a load, read the length at eye level, find the extension, repeat, plot load against extension.
• Sketch a load–extension graph: a straight line through the origin, then a curve for larger loads. Check which quantity is on each axis before you read a gradient.
• Identify the limit of proportionality on a graph: mark the point where the straight line ends. Extended
• Calculate with moments: write “clockwise moment = anticlockwise moment”, then substitute. Keep all distances in the same unit.
• Explain stability with the line of action of the weight: inside the base, the object turns back; outside the base, it topples.
• Describe circular motion with the three cases. Always say which two quantities stay constant. Extended
• A typical 1-mark answer: “The centre of gravity is the point where the whole weight of the object seems to act.”
• A typical 2-mark answer to “Why is a long spanner easier to use?”: “The force is a larger perpendicular distance from the pivot (1), so the same force gives a larger moment (1).”
• A typical 3-mark answer to “Explain why a low, wide object is hard to knock over”: “Its weight acts through its centre of gravity (1). When it tilts, the line of the weight stays inside the base (1), so the weight has a moment that turns it back (1).”
• A typical 2-mark answer to “Define the spring constant”: “The force per unit extension (1). k = F / x (1).” Extended