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- Homework Statement
- A mass at rest on the surface of the earth at latitude λ , experiences a reaction from the earth that consists of a normal component S per unit mass and a tangential frictional component F per unit mass, directed towards a point vertically above the north pole. The earth is assumed to be a uniform sphere of radius R and mass M rotating with an angular velocity ω about its axis. How is the magnitude and direction of the apparent ’acceleration due to gravity’ g , acting on the mass at latitude , related to these forces?

Show that:

$$g^2 = [\frac{GM}{R^2}-R\omega^2cos^2\lambda]^2 + [\frac{1}{2}\omega^2Rsin2\lambda]^2$$

- Relevant Equations
- $$g=-\frac{GMm}{R^2}$$

I began by drawing a diagram and resolving the forces. Since the question asked for 'apparent gravity' I tried to find the normal force.

I started with the equations:

$$\\(\frac{GM}{R^2}-N)sin\lambda-Fsin\lambda=m\omega^2Rcos\lambda$$

$$\\(\frac{GM}{R^2}-N)sin\lambda-Fcos\lambda=0$$

Solving simultaneously, I ended up with:

$$\\N=\frac{GM}{R^2}-R\omega^2cos^2\lambda$$

I started with the equations:

$$\\(\frac{GM}{R^2}-N)sin\lambda-Fsin\lambda=m\omega^2Rcos\lambda$$

$$\\(\frac{GM}{R^2}-N)sin\lambda-Fcos\lambda=0$$

Solving simultaneously, I ended up with:

$$\\N=\frac{GM}{R^2}-R\omega^2cos^2\lambda$$

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