Relative Motion - Earth's rotation and centrifugal acceleration

Problem Statement:

For points A, B, C and D on the surface of the Earth (which we will assume to be a sphere of radius RT), calculate the centrifugal acceleration. The constant angular speed of the earth is ω . Give the results as a function of the givens.

Relative Motion - Earth's rotation and centrifugal acceleration

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Solution:

Centrifugal acceleration

The centrifugal acceleration is given by:

Where ω is the angular velocity of the earth and r is the position vector with respect to O’ (a non-inertial reference frame) of the point where we are calculating it.

First we are going to calculate the magnitude of the centrifugal acceleration as a function of the latitude λ:

We begin by determining the direction of the second cross product in the expression above, ω ⨉ r. We are going to use the right hand rule:

right hand rule

The cross product is orthogonal to both vectors, that is, it is perpendicular to the screen and pointing inwards (in blue in the figure).

The blue symbol written to the right of the cross product, ⊗, is an alternative way to represent a vector perpendicular to the plane created by ω  and r pointing inwards. We will use either one indistinctly.

In the figure above we can also see that the angle between ω and ωr is 900. We will use it to calculate the magnitude of the centrifugal acceleration.

The magnitude of the centrifugal acceleration is given by:

We are going to calculate this magnitude in two steps. First we determine the magnitude of the first cross product knowing that, as we have just seen, the angle between ω and ωr is 900:

Next we are going to calculate the magnitude of the second cross product, ω ⨉ r. The angle between ω and r is 90-λ:

The magnitude of r is the Earth radius, RT. Substituting in the expression above we obtain:

Equation (1) allows us to calculate the magnitude of the centrifugal acceleration at any point on the Earth’s surface. As you can see, the centrifugal acceleration at the poles (λ = ± 900) is zero. On the other hand, centrifugal acceleration reaches its maximum value at the equator (λ = 00).

Next we are going to calculate the centrifugal acceleration vector at each of the points on the Earth’s surface represented in the figure.

Point A: This point is the North Pole; it can be deduced from equation (1) that the centrifugal acceleration at this point is zero because its latitude is 900.


Point B: Equation (1) gives us the magnitude of the centrifugal acceleration at this point. Its direction is determined by using the right hand rule. We had already used it to determine the direction of ω ⨉ r. Now we use it to determine the direction of (ω ⨉(ω ⨉ r)):

right hand rule

From the figure above you can see that the right hand has to be oriented with its fingers parallel to the first vector (ω). Next we close it over the second (ω ⨉ r) which is perpendicular to the screen and pointing inwards. The thumb gives the direction of the cross product.

The minus sign in the expression of the centrifugal acceleration reverses the direction of the vector. Finally, the centrifugal acceleration at point B is given by:


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Point C: Here the latitude is λ but at the southern hemisphere. The magnitude of the centrifugal acceleration is given by equation (1). Its magnitude is given by the right hand rule. The direction of ω  r  is the same as at point B, so is the direction of the centrifugal acceleration. Try it yourself to practice.

The centrifugal acceleration at point C is given by:


Point D: Point D is on the equator. We are going to plot ω and r in this situation, as well as the cross product ω ⨉ rusing the right hand rule:

right hand rule

The blue symbol written to the right of the cross product, ⊙, is an alternative way to represent a vector pointing outwards.

Finally we have to determine the direction of ω ⨉ (ω ⨉ r) using the right hand rule:

right hand rule

The minus sign included in the definition of the centrifugal acceleration reverses the direction of the vector.

Point D is at zero latitude. Using equation (1) to determine the magnitude of the centrifugal acceleration we obtain:

This is where the centrifugal acceleration reaches its maximum value.

Finally we are going to draw the centrifugal acceleration at the different points (vectors are not to scale):

Centrifugal acceleration - latitude

The centrifugal force always points radially outwards.

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