A spring compresses in length by 0.14 in for every 1 lbf of applied force. Determine the mass of an object, in pounds mass, that causes a spring deflection of 1.8 in. The local acceleration of gravity 5 31 ft/s2.
At a certain elevation, the pilot of a balloon has a mass of 120 lb and a weight of 119 lbf. What is the local acceleration of gravity, in ft/s2, at that elevation? If the balloon drifts to another elevation where g 5 32.05 ft/s2, what is her weight, in lbf, and mass, in lb?
The Correct Answer and Explanation is :
Answers:
- Spring Problem:
- The force applied to the spring: 12.86 lbf
- The mass of the object: 13.34 lbm
- Balloon Problem:
- The local acceleration of gravity at the initial elevation: 31.91 ft/s²
- The pilot’s weight at the new elevation: 119.54 lbf
- The pilot’s mass remains the same: 120 lbm (mass is independent of gravity)
Explanation:
Spring Problem
A spring compresses 0.14 inches for every 1 lbf applied. To find the force required for a 1.8-inch deflection, we use:
[
F = \frac{\text{spring deflection}}{\text{deflection per lbf}} = \frac{1.8}{0.14} = 12.86 \text{ lbf}
]
Since weight (( W )) is related to mass (( m )) by gravity:
[
W = m \cdot \frac{g_{\text{local}}}{g_{\text{standard}}}
]
where ( g_{\text{standard}} = 32.174 ) ft/s², and ( g_{\text{local}} = 31 ) ft/s².
[
m = W \times \frac{g_{\text{standard}}}{g_{\text{local}}} = 12.86 \times \frac{32.174}{31} = 13.34 \text{ lbm}
]
Balloon Problem
We know the pilot’s mass is constant at 120 lbm. The weight equation:
[
W = m \cdot \frac{g}{g_{\text{standard}}}
]
Initially, the weight is 119 lbf, so we calculate the local gravity:
[
g_{\text{initial}} = \frac{W \cdot g_{\text{standard}}}{m} = \frac{119 \times 32.174}{120} = 31.91 \text{ ft/s²}
]
At the new elevation where ( g = 32.05 ) ft/s², the new weight is:
[
W_{\text{new}} = m \cdot \frac{g_{\text{new}}}{g_{\text{standard}}} = 120 \times \frac{32.05}{32.174} = 119.54 \text{ lbf}
]
Thus, her weight increases slightly at the new elevation, but her mass remains the same.
These calculations demonstrate how weight varies with gravity, while mass remains constant.