2018 A/L Physics Past Paper · Question 31 (English Medium)

Answer: (5) Figure (5)

What the question asks

We know: $X$ is at rest.

$Z$ moves right at $v$. $Y$ moves left at $\frac{v}{2}$.

We want: the figure that shows the velocity vectors of the oil layers.

Relevant theory

The change of velocity according to the gap between two layers is called the velocity gradient.

This property of a liquid is called viscosity and the forces are viscous forces.

The oil layer touching a plate moves with that plate.

In the steady state the tangential force is the same on every layer.

So the velocity gradient is constant.

So the speed changes in a straight line with distance.

Required equations

$F = -\eta A \frac{dv}{dx}$

$\frac{dv}{dx} = \text{constant}$

Reasoning and calculation

The axes.

Up: the height from $Y$ to $X$.

Across: the velocity of the layer. To the right is positive.

The speeds at the plates:

$0$ at $X$.

$v$ at $Z$, to the right.

$\frac{v}{2}$ at $Y$, to the left.

There are two parts.

The upper layer, between $Z$ and $X$.

The lower layer, between $Y$ and $Z$.

The behaviour changes at plate $Z$.

In the upper layer the speed grows in a straight line from $0$ at $X$ to $v$ at $Z$.

In the lower layer it changes in a straight line from $v$ at $Z$ to $\frac{v}{2}$ to the left at $Y$.

Where is the speed zero in the lower layer?

The total change is from $v$ to $-\frac{v}{2}$, that is $\frac{3v}{2}$.

So the speed is zero two-thirds of the way from $Z$. Below that the arrows point left.

The figure we expect:

No arrow at $X$.

Going down, the arrows grow to the right. They are longest at $Z$.

Then they shrink and turn round near $Y$.

At $Y$ they point left with $\frac{v}{2}$.

Reading the options

Only figure (5) matches this.

The longest arrow points right at $Z$. A small arrow points left at $Y$.

Why the other options fail

This option says: figure (1). The speed at $Y$ is zero.

But $Y$ moves left at $\frac{v}{2}$.

So it is wrong.

This option says: figure (2). A large speed to the right at $Y$.

But $Y$ moves to the left.

This option says: figure (3). In the lower layer the speed does not change in a straight line.

But the velocity gradient is constant. So it must change in a straight line from $v$ at $Z$.

This option says: figure (4). The speed at $Z$ points left.

But $Z$ moves to the right.

Conclusion

$0$ at $X$.

$v$ at $Z$, to the right.

$\frac{v}{2}$ at $Y$, to the left.

Straight lines in between.

The answer is option (5).

Explore

Viscous oil fills the space between the plates. $X$ stays still.

$Z$ is pulled right at $v$. The oil layers nearer $Z$ move faster.

$Y$ is pulled left at $\frac{v}{2}$. Below $Z$ the speed changes in a straight line.

The arrows grow from $X$ to $Z$. Then they shrink and turn left. That is figure (5).