Noninertial/fictitious forces (Movement on a rotating Earth)

[jasker]

Homework Statement

Two trains are traveling east and west on the equator at 50km/h. What is the difference in weight of an apple (mass = 200g) in each train? In which train is it lighter?

Homework Equations

x denotes vector (cross) product, bold denotes a vector quantity. Subscripts rot and in denote the rotating and inertial frames of reference respectively. Ω is the angular velocity of rotating earth. r is the position vector to a point on the equator (magnitude here equal to Re, the radius of earth)

The apparent force due to motion on a rotating Earth is:
Frot = marot
= main - m[2Ω x vrot + Ω x (Ω x r)]
= F + Ffict

The fictitious force Ffict consists of two terms, the Coriolis force and the Centrifugal force:
Coriolis force F = -2mΩ x vrot = -2mΩvrot
Centrifugal force F = -mΩ x (Ω x r) = -mΩ^2 * Re

The Attempt at a Solution

On the train traveling east, the Coriolis force points toward the axis of rotation of the earth, increasing the weight of the apple:
Frot = mg + 2mΩvrot - (mΩ^2 * Re)

On the train traveling west, the Coriolis force points outward from the axis of rotation of the earth, decreasing the weight of the apple:
Frot = mg - 2mΩvrot - (mΩ^2 * Re)

At this point, I am stuck. Is what I have correct so far? How do I calculate the weight of the apple given the two equations above? Do I just plug in these values?
m = 0.2kg
Ω = 2π rad / 86400 sec = 7.27*10^-5 rad/s
Re = 6378.14 km @ equator
vrot = 50 km / h = 13.89 m/s

Thank you!

 

[Mindscrape]

Two comments
1) The problem wants the difference in weights.
2) How is "weight" normally measured?

 

[jasker]

Well I would assume that it would be using the gravitational definition of weight, weight being the force exerted on a body by gravity, so weight would be equal to F = mg, and would be expressed in Newtons (kg * m/s²)

So the weight of the apple at rest would be F = ma_in = mg = 0.2kg * 9.8m/s = 1.96N

On the train traveling east, the weight would be:
F_rot = mg + 2mΩv_rot - (mΩ^2 * Re) = 1.96N + 2(0.4kg)(7.27*10^-5 rad/s)(13.89 m/s) - (0.2kg)(7.27*10^-5 rad/s)²(6378140m) = 1.95406578 N

On the train traveling west, the weight would be:
F_rot = mg - 2mΩv_rot - (mΩ^2 * Re) = 1.96N - 2(0.4kg)(7.27*10^-5 rad/s)(13.89 m/s) - (0.2kg)(7.27*10^-5 rad/s)²(6378140m) = 1.95245009 N

Thus it would be lighter on the train traveling west. Does that all look right?

 

[Mindscrape]

Typically scales measure the normal force, by seeing how much a spring compresses. So really you should have Frot-N=0, but Frot=N, so you're good.

 

[paranormal]

I would first commend the student for their attempt at solving the problem and using the appropriate equations. However, I would also point out that the equations provided may not be necessary for solving this particular problem.

To calculate the difference in weight of the apple in each train, we can use the equation W = mg, where W is the weight, m is the mass, and g is the acceleration due to gravity. In this case, we can assume that g is constant and equal to 9.8 m/s^2.

Using this equation, we can calculate the weight of the apple in the eastbound train:
W = (0.2 kg)(9.8 m/s^2) = 1.96 N

And in the westbound train:
W = (0.2 kg)(9.8 m/s^2) = 1.96 N

Therefore, the difference in weight is 0 N, and the apple will have the same weight in both trains.

It is important to note that the equations provided in the problem may be relevant in other situations, such as when calculating the weight of an object at different latitudes on a rotating Earth. However, for this specific problem, the difference in weight can be easily calculated using the basic formula W = mg.