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Chapter 1: Problem 34

An object whose mass is \(7.8 \mathrm{~kg}\) occupies a volume of \(0.7 \mathrm{~m}^{3}\). Determine its (a) weight, in newtons, and average density, in \(\mathrm{kg} / \mathrm{m}^{3}\), at a location on the earth where \(g=9.55 \mathrm{~m} / \mathrm{s}^{2}\), (b) weight, in newtons, and average density, in \(\mathrm{kg} / \mathrm{m}^{3}\), on the moon where \(g=1.7 \mathrm{~m} / \mathrm{s}^{2}\).

Short Answer

Expert verified
The object's weight and average density on earth are 74.49 N and 11.14 kg/ m^3. Meanwhile, on the moon, the object's weight and density are 13.26 N and 11.14 kg/ m^3 respectively

Step by step solution

01

Calculate Weight on Earth

Calculate the weight of the object on Earth by multiplying the object's mass with the gravitational acceleration on Earth. Using the formula for weight, which is \(W = m \cdot g\), where m is the mass, g is the gravity, we get: \(W_e = 7.8 \, kg \cdot 9.55 \, m/s^2\)
02

Calculate Density on Earth

Calculate the density of the object on Earth using the density formula \( Density = \frac{mass}{volume} \). Substituting the given values, we get: \(Density = \frac{7.8 \, kg}{0.7 \, m^3}\)
03

Calculate Weight on Moon

The weight of the object on the moon is calculated similarly to the weight on Earth but with different gravity. Apply the weight formula with the gravity on the moon: \(W_m = 7.8 \, kg \cdot 1.7 \, m/s^2\)
04

Calculate Density on Moon

The density of the object on the moon remains the same as on the earth, because density is independent of location. For completeness, we use the same equation as in Step 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gravitational Acceleration
Gravitational acceleration is a crucial concept in physics that represents the rate at which an object's velocity increases due to the force of gravity. This acceleration is essential in calculating the weight of objects. The symbol for gravitational acceleration is \(g\). On Earth, \(g\) is approximately \(9.81 \, m/s^2\), but it can vary slightly in different locations due to factors like altitude and Earth's rotation. This exercise uses \( g = 9.55 \, m/s^2 \), showcasing one such variation.

On the Moon, the gravitational acceleration is much less than on Earth, measured at about \(1.7 \, m/s^2\). This decreased gravity affects how much an object weighs, though not its mass. To find an object's weight, we apply the formula \( W = m \cdot g \), where \(m\) is the mass. Thus, an object's weight changes with its location because \(g\) changes, but its mass remains constant.
Density Calculation
Density is a physical property that describes the mass of a substance per unit volume. It provides insight into how packed or spread out matter is within a given space. The formula to calculate density is \( \text{Density} = \frac{\text{mass}}{\text{volume}} \). Density has units of \( \text{kg/m}^3 \).

In the exercise, the mass is \(7.8 \, kg\) and the volume is \(0.7 \, m^3\), so the density is calculated by substituting these values into the formula: \( \frac{7.8 \, kg}{0.7 \, m^3} \approx 11.14 \, kg/m^3 \).

Unlike weight, density is invariant irrespective of location. Whether you're on Earth or the Moon, density remains the same as it depends solely on an object's mass and volume, not on gravitational forces. This makes density a particularly useful attribute when comparing materials.
Mass and Weight Relationships
Understanding the difference between mass and weight is essential in physics. Mass is a measure of the amount of matter in an object, typically measured in kilograms (kg). It remains constant irrespective of where the object is in the universe. On the other hand, weight is the force exerted by gravity on the object鈥檚 mass. It鈥檚 measured in newtons (N) and changes depending on the gravitational pull of the place where the object is located.

To calculate weight, you multiply mass by gravitational acceleration \(g\). For example, an object with a mass of \(7.8 \, kg\) would weigh fewer newtons on the Moon than on Earth due to the Moon's lower gravitational pull. The formula \( W = m \cdot g \) was used in the original exercise to determine this relationship. Even though an object鈥檚 mass remains \(7.8 \, kg\) both on Earth and the Moon, its weight is different due to varying values of \(g\).
  • Mass: Constant everywhere, measure in kilograms.
  • Weight: Variable, depends on gravity, measured in newtons.

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Most popular questions from this chapter

Search online at http://www.osha.gov and print a copy of \(29 \mathrm{CFR} \mathrm{} 1910.212\), General requirements for all machines. With these requirements in mind, identify a machine you have used that had a machine guard to protect the operator or other person in the machine area from hazards. Sketch the machine and label the guarding device, power source, point of operation, and danger zone.

For a vehicle moving at velocity, \(V\), determine the power in \(\mathrm{kW}\) to overcome aerodynamic drag. In general, the drag force, \(F_{d}\), imposed on a vehicle by the surrounding air is given by $$ F_{d}=C_{d} A \frac{1}{2} \rho V^{2} $$ where \(C_{d}\) is a constant for the vehicle called the drag coefficient, \(A\) is the projected frontal area of the vehicle, and \(\rho\) is the air density. For this vehicle, \(C_{d}=0.60, A=10 \mathrm{~m}^{2}, \rho=1.1 \mathrm{~kg} / \mathrm{m}^{2}\), and \(V=100 \mathrm{~km} / \mathrm{h}\).

According to an OSHA director of field programs, a driller was in the process of raising the traveling block and the attached kelly and swivel assembly when an oil drilling rig accident occurred. During an oil drilling operation, an air-chugger winch cable was attached to the kelly pipe as a tag line to prevent it from swinging. Two of the rig hands were monitoring the kelly pipe as it was pulled out of the rat hole. The chain hand picked up the spinning chain and positioned himself near the drawworks drum as he waited for the kelly to be positioned over the hole. When he tossed the end of the spinning chain over his shoulder it became entangled in the fast line cable as the cable was spooled onto the drum. The spinning chain wrapped around the worker's wrist and pulled him into the drum as the chain's slack was drawn up. The spinning chain also struck the worker in the groin area and fractured his leg and severed his femoral artery. The driller stopped the drawworks before the worker was completely pulled into the drum encasement, but the worker was seriously injured. His leg was amputated and injuries to his hand resulted in a permanent disability. The drawworks drum was equipped with metal casing that enclosed the drum on the lower front, top, and sides. However, there was an opening of approximately \(4 \mathrm{ft} \times 3 \mathrm{ft}\) to allow the fast line adequate clearances to spool back and forth onto the drum that has no barrier guard. (See Figure P1.15D.) Apparently, the drawworks drums used on oil well drilling sites were designed and constructed as described above without any barrier guard to protect workers in close proximity to the drawworks drum from the hazard of the ingoing nip point between the moving fast line and the drum. Search the OSHA regulations http://www.osha.gov and specifically review the regulation 29 CFR \(1910.212(\mathrm{a})(1)\), General requirement for all machines. Write a paragraph explaining how this section would apply to the drum. Also, suggest a guard (design) that could have prevented this accident.

A spacecraft component occupies a volume of \(8 \mathrm{ft}^{3}\) and weighs \(25 \mathrm{lb}\) at a location where the acceleration of gravity is \(31.0 \mathrm{ft} / \mathrm{s}^{2}\). Determine its weight, in pounds, and its average density, in \(\mathrm{lbm} / \mathrm{ft}^{3}\), on the moon, where \(g=5.57 \mathrm{ft} / \mathrm{s}^{2}\).

A press delivers a force of \(8000 \mathrm{~N}\) throughout a distance of \(18 \mathrm{~mm}\) each stroke. Select a number of strokes per minute and determine the rate of work output per second. If the press efficiency is \(90 \%\), what average torque must be provided by a 1750 -rpm driving motor?
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