Part A1: Consider The Arm Of A Balance; A Mass Hangs At Poin

Part A1consider The Arm Of A Balance Ab A Massmhangs At Point A O

Part A1consider The Arm Of A Balance Ab A Massmhangs At Point A O

Consider the arm of a balance AB. A mass m hangs at point A on a string of length y. The weight of that mass is W. A second mass m hangs at point B on a string of length y + h. The weight of that mass is W'. The strings are assumed to have negligible mass. The problem is to illustrate the scenario and demonstrate that the difference in weights associated with the fact that W' is closer to the Earth than W is given by the relation:

W' - W = (8/3) G m h / ρ_m,

where ρ_m is the mean density of the Earth, and the Earth is modeled as a perfect sphere. This involves analyzing how the gravitational force varies with height and how the gravitational field strength decreases with altitude, based on Earth's density distribution.

Paper For Above instruction

The problem revolves around understanding the variation of gravitational acceleration at different points along the balance arm and relating this to the difference in weights W and W'. When a mass is placed at a certain height above Earth's surface, the gravitational acceleration slightly decreases due to the inverse-square law. Assuming the Earth as a perfect sphere with uniform density simplifies the calculations significantly.

To derive the relation, we start with Newton’s law of universal gravitation and integrate over the Earth's volume considering its mean density ρ_m. For a small height h above the Earth's surface, the fractional change in gravitational acceleration g is approximately given by:

Δg / g ≈ -h / R,

where R is Earth's radius. Considering the gravity variation and the smallness of h relative to R, the difference in weights W' and W corresponds to the change in g at different distances from Earth's center.

This leads to an expression for the difference W' - W in terms of G, m, h, and ρ_m, specifically yielding:

W' - W = (8/3) G m h / ρ_m

This relation illustrates how the slight decrease in gravitational acceleration at higher elevation affects the weight measured on the balance and simplifies as the ratio of the height difference to Earth's parameters.

References

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