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Chapter 1: Problem 72

Give the derived SI units for each of the following quantities in base SI units: (a) acceleration \(=\) distance \(/\) time \(^{2}\) (b) force \(=\) mass \(\times\) acceleration (c) work \(=\) force \(\times\) distance (d) pressure \(=\) force/area (e) power = work/time (f) velocity = distance/time (g) energy \(=\) mass \(\times(\text { velocity })^{2}\)

Short Answer

Expert verified
(a) m/s虏, (b) N, (c) J, (d) Pa, (e) W, (f) m/s, (g) J.

Step by step solution

01

Analyzing Acceleration

Acceleration is given by the formula \( a = \frac{d}{t^2} \), where \( d \) is distance and \( t \) is time. In base SI units, distance is measured in meters (m) and time is in seconds (s). Thus, the derived SI unit for acceleration is \( \frac{\text{m}}{\text{s}^2} \), or \( \text{m/s}^2 \).
02

Determining Force Units

Force is calculated using \( F = m \cdot a \), where \( m \) is mass and \( a \) is acceleration. Mass in SI units is in kilograms (kg) and acceleration is in meters per second squared (m/s虏). Therefore, the derived unit for force is \( \text{kg}\cdot\text{m/s}^2 \), known as a newton (N).
03

Calculating Work Units

Work is given by \( W = F \cdot d \), where \( F \) is force and \( d \) is distance. The derived unit for force is newtons (N) and distance is in meters (m). Thus, work is measured in newton-meters (N鈰卪), also known as a joule (J).
04

Establishing Pressure Units

Pressure is defined as \( P = \frac{F}{A} \), where \( F \) is force and \( A \) is area. The force is measured in newtons (N) and area is in square meters (m虏). Thus, the derived SI unit for pressure is \( \frac{\text{N}}{\text{m}^2} \), which is called a pascal (Pa).
05

Deriving Power Units

Power is defined by \( P = \frac{W}{t} \), where \( W \) is work and \( t \) is time. Work is measured in joules (J) and time in seconds (s). Therefore, the unit for power is \( \frac{\text{J}}{\text{s}} \), known as a watt (W).
06

Analyzing Velocity Units

Velocity is given by \( v = \frac{d}{t} \), with distance \( d \) in meters (m) and time \( t \) in seconds (s). Thus, the unit for velocity is meters per second (m/s).
07

Calculating Energy Units

Energy is calculated using \( E = m \cdot v^2 \). Mass \( m \) is in kilograms (kg) and velocity \( v \) is in meters per second (m/s), squared in this case. Thus, the derived unit for energy is \( \text{kg}\cdot(\text{m/s})^2 \), which simplifies to a joule (J).

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Key Concepts

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

Acceleration
Acceleration is a measure of how quickly something speeds up or slows down. It involves a change in velocity over time. To calculate acceleration, we use the formula \( a = \frac{d}{t^2} \), where \( d \) is distance and \( t \) is time.
  • Distance is measured in meters \( (\text{m}) \).
  • Time is measured in seconds \( (\text{s}) \).
Therefore, the unit of acceleration is meters per second squared, expressed as \( \text{m/s}^2 \). This tells us how much the speed changes every second.
Force
Force describes the push or pull on an object, affecting its motion. It is calculated using the formula \( F = m \cdot a \), where:
  • \( m \) is mass in kilograms \( (\text{kg}) \).
  • \( a \) is acceleration in meters per second squared \( (\text{m/s}^2) \).
The resulting unit for force is \( \text{kg} \cdot \text{m/s}^2 \), known as a newton \( (\text{N}) \). One newton is the amount of force required to accelerate a one-kilogram mass by one meter per second squared.
Work
Work refers to the energy transferred when a force moves an object over a distance. The equation for work is \( W = F \cdot d \), where:
  • \( F \) is force in newtons \( (\text{N}) \).
  • \( d \) is distance in meters \( (\text{m}) \).
The derived unit for work is the newton-meter, or joule \( (\text{J}) \). This unit indicates how much energy is used to move an object.
Pressure
Pressure is the amount of force applied per unit area. It is defined by the formula \( P = \frac{F}{A} \), where:
  • \( F \) is force in newtons \( (\text{N}) \).
  • \( A \) is area in square meters \( (\text{m}^2) \).
The resulting unit of pressure is newtons per square meter, known as a pascal \( (\text{Pa}) \). This tells us how force is spread over an area.
Power
Power measures how quickly work is done or energy is transferred. It is calculated using \( P = \frac{W}{t} \), which breaks down as follows:
  • \( W \) is work in joules \( (\text{J}) \).
  • \( t \) is time in seconds \( (\text{s}) \).
The power unit is joules per second, more commonly known as watts \( (\text{W}) \). A watt represents one joule of work done in one second.
Velocity
Velocity describes the speed of an object in a particular direction. It is expressed by \( v = \frac{d}{t} \), where:
  • \( d \) represents distance in meters \( (\text{m}) \).
  • \( t \) is time in seconds \( (\text{s}) \).
The unit for velocity is meters per second \( (\text{m/s}) \). This indicates both how fast and in which direction an object is moving.
Energy
Energy is the capacity to do work. It is determined with \( E = m \cdot v^2 \), involving:
  • \( m \) as mass in kilograms \( (\text{kg}) \).
  • \( v \) as velocity in meters per second \( (\text{m/s}) \).
Squaring velocity means multiplying it by itself, \( (\text{m/s})^2 \). The derived SI unit for energy is the joule \( (\text{J}) \), indicating the ability of an object to perform work.

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Most popular questions from this chapter

(a) Three spheres of equal size are composed of aluminum \(\left(\right.\) density \(\left.=2.70 \mathrm{~g} / \mathrm{cm}^{3}\right),\) silver \(\left(\right.\) density \(\left.=10.49 \mathrm{~g} / \mathrm{cm}^{3}\right)\) and nickel (density \(\left.=8.90 \mathrm{~g} / \mathrm{cm}^{3}\right) .\) List the spheres from lightest to heaviest. (b) Three cubes of equal mass are composed of gold (density \(\left.=19.32 \mathrm{~g} / \mathrm{cm}^{3}\right),\) platinum (density \(\left.=21.45 \mathrm{~g} / \mathrm{cm}^{3}\right),\) and lead \(\left(\right.\) density \(\left.=11.35 \mathrm{~g} / \mathrm{cm}^{3}\right)\) List the cubes from smallest to largest.

Classify each of the following as a pure substance or a mixture. If a mixture, indicate whether it is homogeneous or heterogeneous: (a) milk, (b) beer, (c) diamond, (d) mayonnaise.

A \(10.0 \mathrm{~g}\) block of gold is hammered into a thin gold sheet which has an area of \(150 \mathrm{~cm}^{2}\). Given the density of gold is \(19.3 \mathrm{~g} / \mathrm{cm}^{3}\), what is the approximate thickness of the gold sheet in millimeters?

The distance from Earth to the Moon is approximately \(240,000 \mathrm{mi} .\) (a) What is this distance in meters? (b) The peregrine falcon has been measured as traveling up to \(350 \mathrm{~km} /\) hr in a dive. If this falcon could fly to the Moon at this speed, how many seconds would it take? (c) The speed of light is \(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}\). How long does it take for light to travel from Earth to the Moon and back again? (d) Earth travels around the Sun at an average speed of \(29.783 \mathrm{~km} / \mathrm{s}\). Convert this speed to miles per hour.

The density of air at ordinary atmospheric pressure and \(25^{\circ} \mathrm{C}\) is \(1.19 \mathrm{~g} / \mathrm{L}\). What is the mass, in kilograms, of the air in a room that measures \(4.5 \mathrm{~m} \times 5.0 \mathrm{~m} \times 2.5 \mathrm{~m}\) ?
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