Question

At \(45^{\circ}\) latitude, the gravitational acceleration as a function of elevation \(z\) above sea level is given by \(g=a-b z\) where \(a=9.807 \mathrm{m} / \mathrm{s}^{2}\) and \(b=3.32 \times 10^{-6} \mathrm{s}^{-2}\). Determine the height above sea level where the weight of an object will decrease by 0.3 percent.

Short answer

Answer: The height above sea level where the weight of an object will decrease by 0.3% is approximately 903.61 meters.

Step-by-step solution

Understand the relationship between weight and gravitational acceleration

The weight of an object is given by the formula \(W = mg\), where \(W\) is the weight, \(m\) is the mass of the object, and \(g\) is the gravitational acceleration. Since the gravitational acceleration changes with elevation, we can think of the weight of an object as a function of elevation \(z\), so we have \(W(z) = m (a - bz)\).

Calculate the final weight

We are given that the weight of the object decreases by 0.3%, which implies that the final weight will be 99.7% of the initial weight. Let \(W_0\) be the initial weight of the object at sea level and \(W_f\) be the final weight at some elevation \(z\). Then, we have \(W_f = 0.997W_0\).

Set up the equation relating initial and final weights

We know that the initial weight will be given by \(W_0 = m(a - b(0)) = ma\), and the final weight will be given by \(W_f = m(a - bz)\). Using the relationship found in Step 2, we now have: \(0.997W_0 = W_f = m(a - bz)\).

Solve for the elevation, z

Since we are interested in the elevation \(z\) where the weight will decrease by 0.3%, we can eliminate the mass \(m\) and the initial weight \(W_0\) from the equation: \(0.997(ma) = m(a - bz)\) Divide both sides by \(m\) and \(a\), we get: \(0.997 = 1 - bz\) Now, plug in \(a = 9.807 \frac{m}{s^2}\) and \(b = 3.32 \times 10^{-6} s^{-2}\) and solve for \(z\): \(0.003 = (3.32 \times 10^{-6} s^{-2})z\) Thus: \(z = \frac{0.003}{3.32 \times 10^{-6} s^{-2}}\)

Calculate the height above sea level

Now, perform the calculation: \(z = \frac{0.003}{3.32 \times 10^{-6} s^{-2}} = 903.614 m\) Therefore, the height above sea level where the weight of an object will decrease by 0.3% is approximately 903.61 meters.

Key concepts

Weight of an Object

The term 'weight' describes the force exerted by gravity on an object, and is calculated using the formula \( W = mg \), where \( W \) is the weight, \( m \) is the mass of the object, and \( g \) represents the gravitational acceleration. This relationship means that weight is directly proportional to both the mass of the object and the acceleration due to gravity at that specific location.

In simple terms, weight is not a constant value for an object and can change depending on where it is measured due to the variability of \( g \). It's a common misconception that mass and weight are the same, but mass is a measure of how much matter an object contains and remains constant regardless of location, while weight can change if the force of gravity changes.

Elevation and Gravity Relationship

Gravity is not uniform across the Earth's surface. The force of gravity that an object experiences changes slightly with elevation. The farther you are from the Earth's center, the weaker the gravitational pull you experience. This is because the force of gravity between two masses is inversely related to the square of the distance between their centers, as per Newton's law of universal gravitation.

The formula \( g = a - bz \) reveals how gravitational acceleration, \( g \), decreases as the elevation, \( z \), increases. Here, \( a \) and \( b \) are constants; \( a \) represents the gravitational acceleration at sea level, and \( b \) is the rate at which gravitational acceleration decreases with increasing elevation. Students should understand that as they ascend, say from sea level to the top of a mountain, gravity's pull weakens ever so slightly, leading to a decrease in weight.

Weight Decrease with Altitude

To understand how weight decreases with altitude, it’s vital to grasp that this decrease is due to the reduction in gravitational acceleration as elevation increases. As you move higher from the Earth’s surface, the distance between you and the Earth’s center increases, causing a decrease in the gravitational force and, hence, a decrease in weight.

It’s essential to note that while the change in weight is noticeable at significant elevations, it is minuscule at smaller ones. The exercise presented illustrates a weight decrease of 0.3% at an elevation of approximately 903.61 meters, which might not be perceptible in everyday settings. Nonetheless, understanding this concept is crucial in fields like aviation and space exploration where accurate measurements of weight are pivotal due to the considerable altitudes involved.