2018 A/L Physics Past Paper · Question 12 (English Medium)

Answer: (4) Diagram (4)

What the question asks

We know: five forces A, B, C, D and E of the same size $F$ act on a body.

We want: which diagram shows the direction of their resultant?

Relevant theory

If two forces are represented in magnitude and direction by two adjacent sides of a parallelogram, their resultant will be represented given by the corresponding diagonal in magnitude and direction.

Applying it to this question

From the figure, each force is at 45° to the axes.

A: up and to the right.

B: up and to the right.

C: down and to the right.

D: down and to the right.

E: down and to the left.

Split each force into two parts: across ($x$) and up or down ($y$).

Right is $+$. Up is $+$.

At 45°, each part is $F/\sqrt{2}$. We count in these units, so each part is $+1$ or $-1$.

Reasoning and calculation

A and E

$(0, 0)$

A is $(+1, +1)$.

E is $(-1, -1)$.

They are the same size and point opposite ways.

So they cancel.

B

$(+1, +1)$

B points up and to the right.

So B is $(+1, +1)$.

C and D

$(+2, −2)$

C and D both point down and to the right.

Each one is $(+1, -1)$.

Together they are $(+2, -2)$.

Now add the parts.

Across:

$$\Sigma x = 0 + 1 + 2 = +3$$

Up or down:

$$\Sigma y = 0 + 1 - 2 = -1$$

So the resultant is $(+3, -1)$, in units of $F/\sqrt{2}$.

$+3$ across means it points to the right.

$-1$ means it points a little down.

It is about $18^\circ$ below the horizontal.

Why the other options fail

This option says: diagram (1), an arrow going up and to the right.

But the resultant goes down, because $\Sigma y = -1$.

So it is wrong.

This option says: diagram (2), an arrow going steeply up.

But the resultant goes to the right and a little down.

This option says: diagram (3), an arrow going down and to the left.

But the resultant goes to the right, because $\Sigma x = +3$.

This option says: diagram (5), an arrow going up and to the left.

But the resultant goes to the right and down. That is the opposite way.

Checks

A quick check.

A and E are on the same line, the same size, opposite ways. They cancel.

So only B, C and D are left. Two of them point down, one points up.

So the resultant must point down and to the right.

Conclusion

The resultant is $(+3, -1)$: to the right and a little down.

The answer is diagram (4).