A plumb bob (a mass m hanging on a string) is deflected from the vertical by an angle θ due to a massive mountain nearby (Fig. 6–37). .

(c) Estimate the angle θ of the plumb bob if it is 5 km from the center of Mt. Everest. <IMAGE>

Verified step by step guidance

1. Identify the forces acting on the plumb bob: the gravitational force due to Earth (F_g) and the gravitational force due to the mountain (F_m).
2. Understand that the deflection angle θ is caused by the horizontal component of the gravitational force exerted by the mountain. The vertical component of Earth's gravity keeps the plumb bob hanging, while the horizontal component due to the mountain causes the deflection.
3. Use the tangent function to relate the forces and the angle: \( \tan(\theta) = \frac{F_m}{F_g} \). Here, \( F_m \) is the horizontal force due to the mountain and \( F_g \) is the vertical gravitational force due to Earth.
4. Calculate the gravitational force exerted by the mountain using Newton's law of universal gravitation: \( F_m = G \frac{mM}{r^2} \), where G is the gravitational constant, m is the mass of the plumb bob, M is the mass of the mountain, and r is the distance from the center of the mountain to the plumb bob.
5. Substitute the values and solve for θ using the equation \( \theta = \tan^{-1}\left(\frac{F_m}{F_g}\right) \).