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). Estimate the angle θ of the plumb bob if it is 5 km from the center of Mt. Everest.

Verified step by step guidance

1. Identify the forces acting on the plumb bob: The gravitational force due to the Earth (Fₑ) acts vertically downward, and the gravitational force due to the mountain (Fₘ) acts horizontally toward the mountain. The resultant force causes the plumb bob to deflect at an angle θ from the vertical.
2. Express the forces mathematically: The gravitational force due to the Earth is Fₑ = m * g, where g is the acceleration due to gravity. The gravitational force due to the mountain can be approximated using Newton's law of gravitation: Fₘ = G * (m * Mₘ) / r², where G is the gravitational constant, Mₘ is the mass of the mountain, and r is the distance from the plumb bob to the mountain.
3. Relate the forces to the angle θ: The angle θ can be determined using the tangent function, as tan(θ) = Fₘ / Fₑ. Substituting the expressions for Fₘ and Fₑ, we get tan(θ) = [G * Mₘ / r²] / g.
4. Simplify the expression for θ: Solve for θ by taking the arctangent of both sides, θ = arctan([G * Mₘ] / [r² * g]).
5. Substitute the known values: Use the given distance r = 5 km (convert to meters), the gravitational constant G = 6.674 × 10⁻¹¹ N·m²/kg², the approximate mass of Mt. Everest Mₘ (which can be estimated based on its volume and density), and g = 9.8 m/s² to calculate θ. Ensure all units are consistent before performing the calculation.

Key Concepts

Gravitational Force

Gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. It states that every point mass attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. In this scenario, the plumb bob experiences the gravitational pull from both the Earth and the nearby mountain, affecting its equilibrium position.

Deflection Angle

The deflection angle θ represents the angle by which the plumb bob is displaced from the vertical due to the gravitational influence of the nearby mountain. This angle can be calculated using the balance of forces acting on the bob, where the tension in the string and the gravitational forces from both the Earth and the mountain create a resultant force that causes the bob to deviate from its original vertical position.

Distance and Gravitational Influence

The distance from the plumb bob to the center of the mountain is crucial in determining the gravitational influence exerted by the mountain. As the distance increases, the gravitational force decreases according to the inverse square law. In this case, the distance of 5 km from the center of Mt. Everest will influence the magnitude of the gravitational pull on the plumb bob, thereby affecting the angle of deflection θ.