Circular Motion at a Latitude | CAIE 9702 Paper 4 Notes

Paper 4

Physics9702

The path radius at a latitude

The Earth turns around its spin axis. A point away from the Equator moves in a smaller circle around this axis.
$\lambda$ is the angle north or south of the Equator.
Use the horizontal part of the Earth's radius to find the circular path radius.
The inward direction is towards the spin axis. It is not towards the Earth's centre.

r=R\cos\lambda

PATH

Find the radius around the spin axis

This is smaller than the Earth's radius.

R = Earth radiusr = path radiusr = R cos λ

Use the distance from the spin axis, not the distance from the Earth's centre.

Worked example

A student at latitude 52.2°

A student of mass 58.6 kg is at latitude 52.2°. The Earth has radius $6.37\times10^6\text{ m}$ and period 24 hours. Find the path radius, speed and inward resultant force (9702/42/O/N/25 Q1(a)-Q1(b)(i)).

Find the path radius.
$r=R\cos\lambda$
$r=6.37\times10^6\cos52.2^\circ$
$r=3.90\times10^6\text{ m}$
Change the period into seconds, then find the speed.
$T=24\times60\times60$
$T=86400\text{ s}$
$v=\dfrac{2\pi r}{T}=280\text{ m s}^{-1}$
Find the inward resultant force.
$F=\dfrac{mv^2}{r}=1.2\text{ N}$

Answer

r=3.90\times10^6\text{ m},\quad v=280\text{ m s}^{-1},\quad F=1.2\text{ N}

The two real forces

The student has two real forces: weight and the total ground contact force.
Their vector sum is the small resultant towards the spin axis.

The resultant is not a third force. Move one real-force arrow head to tail with the other; the start-to-finish arrow is the resultant.

Weight points towards the Earth's centre.
The total ground contact force is tilted from the normal line towards the spin axis. It is not exactly at 90° to the ground.
The ground contact force is the total of normal contact force and friction.
To add the two real forces, move one arrow so that it starts at the end of the other arrow. Do not turn it or change its length.

Your turn9702/42/O/N/25 Q1(b)(ii)-Q1(b)(iii)

State the direction of the resultant. Then name the two real forces on the student.

Show worked answer

The resultant points towards the spin axis. The real forces are weight and the contact force from the ground.

Common mistake

Do not draw a third force called centripetal force. The inward resultant is the vector sum of weight and the ground contact force.