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Chapter 1: Problem 23

Calculate the force necessary to accelerate a \(20,000-\mathrm{lbm}\) rocket vertically upward at the rate of \(100 \mathrm{ft} / \mathrm{s}^{2}\). Assume \(g=32.2 \mathrm{ft} / \mathrm{s}^{2}\).

Short Answer

Expert verified
62,112 lb

Step by step solution

01

Convert Mass from Pounds-mass to Slugs

First, convert the rocket's mass from pounds-mass (\text{lbm}) to slugs. The conversion factor is 1 slug = 32.2 \text{lbm}. Use the formula: \[ \text{Mass in slugs} = \frac{\text{Mass in lbm}}{g} \] where \( g = 32.2 \text{ ft/} \text{s}^{2} \). \[ \text{Mass in slugs} = \frac{20,000 \text{ lbm}}{32.2 \text{ ft/} \text{s}^{2}} = 621.12 \text{ slugs} \]
02

Calculate the Force

Next, calculate the force required to accelerate the rocket using Newton's second law, \( F = ma \), where \( m \) is the mass in slugs and \( a \) is the acceleration. Substitute \( m = 621.12 \text{ slugs} \) and \( a = 100 \text{ ft/} \text{s}^{2} \). \[ F = (621.12 \text{ slugs}) \times (100 \text{ ft/} \text{s}^{2}) = 62,112 \text{ lb} \]

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Key Concepts

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

Newton's second law
Newton's second law is crucial in understanding how force, mass, and acceleration interact with each other. This law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. The mathematical representation of Newton's second law is given by: \( F = ma \). Here, \( F \) stands for force, \( m \) for mass, and \( a \) for acceleration. This simple equation allows us to calculate the required force if we know the mass and the desired acceleration. In our example, we calculated the force needed to accelerate a 20,000-lbm rocket at 100 ft/s虏. This concept tells us that heavier objects need more force to achieve the same acceleration as lighter objects. It is a fundamental principle in physics and has vast applications in everyday life and engineering projects.
mass and weight conversion
Mass and weight often get confused, but they are different concepts. Mass is the amount of matter in an object and is usually measured in kilograms or pounds-mass (lbm). Weight, on the other hand, is the force exerted by gravity on that mass. Weight is measured in newtons or pounds-force (lbf). The crucial point here is that mass remains constant regardless of location, but weight can change with the strength of the gravitational field. In our example, we converted the rocket's mass from pounds-mass (\text{lbm}) to slugs for easier calculation. One slug is equivalent to 32.2 lbm. So, we used the formula: \( \frac{\text{Mass in lbm}}{32.2 \text{ ft/} \text{s}^{2}} \) to find the mass in slugs. This conversion is important because it allows us to use the same units in physics equations, making calculations more straightforward.
acceleration in physics
Acceleration is the rate of change of velocity of an object. It is a vector quantity, meaning it has both magnitude and direction. In physics, acceleration can be due to a change in speed, direction, or both. The standard unit of acceleration is meters per second squared (m/s虏) or feet per second squared (ft/s虏) in the imperial system. To calculate the force needed to accelerate an object, we need to know the acceleration: \( F = ma \). In the given exercise, the rocket is accelerating upwards at 100 ft/s虏. By knowing this, along with the mass of the rocket in slugs, we can determine the required force to achieve this acceleration. Understanding acceleration helps in solving real-world problems, from designing safe vehicles to launching rockets.

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Most popular questions from this chapter

A bell jar \(250 \mathrm{~mm}\) in diameter sits on a flat plate and is evacuated until a vacuum of \(700 \mathrm{mmHg}\) exists. The local barometer reads \(760 \mathrm{~mm}\) mercury. Find the absolute pressure inside the jar, and determine the force required to lift the jar off the plate. Neglect the weight of the jar.

A temperature of a body is measured to be \(26^{\circ} \mathrm{C}\). Determine the temperature in \({ }^{\circ} \mathrm{R}, \mathrm{K}\), and \({ }^{\circ} \mathrm{F}\).

The weight of a 10 -lb mass is measured at a location where \(g=32.1 \mathrm{ft} / \mathrm{s}^{2}\) on a spring scale originally calibrated in a region where \(g=32.3 \mathrm{ft} / \mathrm{s}^{2}\). What will be the reading?

The gas in a cubical volume with sides at different temperatures is suddenly isolated with reference to transfer of mass and energy. Is this system in thermodynamic equilibrium? Why or why not? It is not in thermodynamic equilibrium. If the sides of the container are at different temperatures, the temperature is not uniform over the entire volume, a requirement of thermodynamic equilibrium. After a period of time elapses, if the sides all approach the same temperature, equilibrium would eventually be attained.

When a body is accelerated under water, some of the water is also accelerated. This makes the body appear to have a larger mass than it actually has. For a sphere at rest this added mass is equal to the mass of one-half of the displaced water. Calculate the initial force necessary to accelerate a \(10-\mathrm{kg}, 300-\mathrm{mm}\)-diameter sphere which is at rest under water at the rate of \(10 \mathrm{~m} / \mathrm{s}^{2}\) in the horizontal direction. Use \(\rho_{\mathrm{H}_{2} \mathrm{O}}=1000 \mathrm{~kg} / \mathrm{m}^{3}\). The added mass is one-half of the mass of the displaced water: m_{\text {added }}=\frac{1}{2}\left(\frac{4}{3} \pi r^{3} \rho_{\mathrm{H}_{2} \mathrm{O}}\right)=\left(\frac{1}{2}\right)\left(\frac{4}{3}\right)(\pi)\left(\frac{0.3}{2}\right)^{3}(1000)=7.069 \mathrm{~kg} The apparent mass of the body is then \(m_{\text {apparent }}=m+m_{\text {added }}=10+7.069=17.069 \mathrm{~kg}\). The initial force needed to accelerate this body from rest is calculated to be F=m a=(17.069)(10)=170.7 \mathrm{~N}
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