On a force diagram, what does each arrow show, and what is written beside it?

Each arrow shows a force acting on the particle, and the direction the arrow points is the direction in which that force acts. The magnitude of the force, in newtons, is written next to the arrow.

Define resultant force .

The resultant force on a particle is the sum of all the forces acting on it. It is the single force that would have exactly the same effect as all of those forces acting together.

Define equilibrium , for a particle.

A particle is in equilibrium when the resultant force acting on it is zero . Its forces are then described as balanced , and any equation you write sets their total equal to zero.

True or False? If the resultant force on a particle is zero, the particle must be at rest.

False. A zero resultant force means zero acceleration , which is not the same as zero velocity. By Newton's first law such a particle either stays at rest or carries on moving with constant velocity .

True or False? The two perpendicular directions used in a two-dimensional force problem do not have to be horizontal and vertical.

True. The only requirement is that the two directions are at right angles to each other. For a particle on a slope it is usually much easier to take them parallel to the slope and perpendicular to it.

Several forces acting on a particle in equilibrium are drawn nose to tail. What shape do they form, and why?

They form a closed polygon , with the last arrow finishing exactly where the first one started. Drawing vectors nose to tail makes their resultant the arrow from the very start to the very end, and in equilibrium that resultant is zero , so it has no length at all.

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Forces & Equilibrium (AQA AS Maths: Mechanics): Flashcards

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On a force diagram, what does each arrow show, and what is written beside it?

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On a force diagram, what does each arrow show, and what is written beside it?
Each arrow shows a force acting on the particle, and the direction the arrow points is the direction in which that force acts.
The magnitude of the force, in newtons, is written next to the arrow.
Fill in the missing force labels:
$______$ N is used for both tension and thrust, $______$ N for friction, and $______$ N for the normal reaction.
$T$ N is used for both tension and thrust, $F$ N for friction, and $R$ N for the normal reaction.
Weight is labelled $W$ N, or written directly as $m g$ N.
A question involves a force whose size you do not yet know. What should you do with it on the force diagram?
Draw its arrow in the direction it acts and label it with a letter, such as $F_{1}$ N or $T$ N.
An unknown force still acts on the particle, so leaving it off would leave it out of the equations too, and it is often the very thing you are asked to find.
True or False?
An acceleration marked on a force diagram is one of the forces acting on the particle.
False.
An acceleration is often marked on a force diagram with its own arrow, labelled $a {m s}^{- 2}$ , to show which way the particle is speeding up.
It is not a force: it is the effect the forces produce, so it must never be included when the forces are added up.
A particle of mass 2 kg is held by three strings, and the diagram marks only the three tensions. Which force is missing, and why must it be added?
The weight of the particle, $2 g$ N, acting vertically downwards.
Every object with mass has a weight acting on it whether or not a diagram marks it, and leaving it out would make the vertical equation wrong.
One object in a question has a weight of 52 N and another has a mass of $m$ kg. What downward force goes on the diagram for each?
For the first, 52 N: a weight is already a force in newtons, so it is not multiplied by $g$ .
For the second, $m g$ N: a mass in kilograms has to be multiplied by $g$ to turn it into a force.
Define resultant force.
The resultant force on a particle is the sum of all the forces acting on it.
It is the single force that would have exactly the same effect as all of those forces acting together.
Forces of 6 N and 3 N act to the right on a particle, and a force of 12 N acts to the left. Find the resultant force.
The resultant force is 3 N to the left.
Taking rightwards as positive, the forces add to $6 + 3 - 12 = - 3$ , and the negative sign shows that the resultant acts to the left.
Define equilibrium, for a particle.
A particle is in equilibrium when the resultant force acting on it is zero.
Its forces are then described as balanced, and any equation you write sets their total equal to zero.
Two forces act to the right on a particle and only one acts to the left. Can the particle still be in equilibrium?
Yes. Equilibrium depends on the total force each way, not on how many forces there are on each side.
A single 12 N force to the left balances forces of 7 N and 5 N to the right, because $7 + 5 = 12$ .
Fill in the two missing words:
A resultant force that is not zero is described as $______$ , and the particle it acts on will $______$ .
A resultant force that is not zero is described as unbalanced, and the particle it acts on will accelerate.
An unbalanced force is one that is not cancelled out by another force acting in the opposite direction.
A shop sign of mass $(5 x - 7)$ kg hangs in equilibrium from two strings. What must be done with the mass before the equilibrium equation is written?
It must be turned into a weight by multiplying by $g$ , giving a downward force of $(5 x - 7) g$ N.
An equilibrium equation balances forces, so every term in it has to be in newtons, and a mass in kilograms is not a force.
True or False?
If the resultant force on a particle is zero, the particle must be at rest.
False.
A zero resultant force means zero acceleration, which is not the same as zero velocity.
By Newton's first law such a particle either stays at rest or carries on moving with constant velocity.
True or False?
The two perpendicular directions used in a two-dimensional force problem do not have to be horizontal and vertical.
True.
The only requirement is that the two directions are at right angles to each other.
For a particle on a slope it is usually much easier to take them parallel to the slope and perpendicular to it.
Several forces acting on a particle in equilibrium are drawn nose to tail. What shape do they form, and why?
They form a closed polygon, with the last arrow finishing exactly where the first one started.
Drawing vectors nose to tail makes their resultant the arrow from the very start to the very end, and in equilibrium that resultant is zero, so it has no length at all.
A particle of weight $2 g$ N is held in equilibrium by three strings, pulling $(4 x + 2)$ N upwards, $(x + 7)$ N to the left and $F$ N to the right. Write the two equilibrium equations.
The two equations are $F - (x + 7) = 0$ horizontally and $(4 x + 2) - 2 g = 0$ vertically.
Start with the vertical equation, because $x$ is its only unknown, and then substitute the value of $x$ into the horizontal one.
A particle has forces of 8 N to the right, 8 N to the left, 5 N up and 3 N down acting on it. Is it in equilibrium?
No. The horizontal forces balance, but vertically there is a resultant of $5 - 3 = 2$ N upwards.
Balancing in one direction is not enough: a resultant that is not zero in either direction leaves the particle unbalanced, and it will accelerate upwards.
A force is $(- 2 i - 3 j) N$ . Find its magnitude.
The magnitude is $\sqrt{13} N$ , which is $3.61 N$ to 3 significant figures.
The two components are perpendicular, so Pythagoras' theorem gives:
$| F | = \sqrt{(- 2)^{2} + (- 3)^{2}} = \sqrt{13} N$
The signs disappear when the components are squared, so a magnitude is never negative.
How is the direction of a force in two dimensions usually given, and what is it measured from?
The direction is given as an angle, usually in degrees, measured anticlockwise from the positive horizontal direction.
So a direction of $216.9 °$ describes a force pointing down and to the left.
True or False?
A force given as $(3 i + 8 j) N$ tells you its magnitude and its direction straight away.
False.
Component form separates a force into its horizontal and vertical parts, so neither the magnitude nor the direction is shown directly.
The magnitude comes from Pythagoras' theorem, and the direction from a sketch and trigonometry.
A force is $(- 8 i - 6 j) N$ . Why does putting $\tan^{- 1} (\frac{- 6}{- 8})$ into a calculator not give its direction?
A calculator returns an angle between $- 90 °$ and $90 °$ , so it cannot distinguish between two opposite directions.
Sketch the components instead, find the acute angle from the sizes of the components, then adjust it. Here $\tan^{- 1} (\frac{6}{8}) = 36.9 °$ , and the force points down and to the left, so the direction is $180 ° + 36.9 ° = 216.9 °$ , measured anticlockwise from the positive horizontal direction.
What does the resultant force on a particle in equilibrium look like when the forces are written in component form?
The resultant force is the zero vector, written $0$ N.
In component form that is $0 i + 0 j$ , or as a column vector $(\begin{matrix} 0 \\ 0 \end{matrix})$ . Both components have to be zero separately, which is the vector form of the two equilibrium equations.
Two forces act on a particle and their resultant is $(- 2 i - 3 j) N$ . What third force would bring the particle into equilibrium?
The third force is $(2 i + 3 j) N$ .
All three forces must add to the zero vector, so the third force has both components of the existing resultant reversed.

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