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Chapter 1: Problem 72

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

Short Answer

Expert verified
The derived SI units for each quantity are as follows: (a) Acceleration: \[\frac{m}{s^2}\] (b) Force: \[kg\cdot\frac{m}{s^2}\] (c) Work: \[kg\cdot\frac{m^2}{s^2}\] (d) Pressure: \[\frac{kg}{m\cdot s^2}\] (e) Power: \[\frac{kg\cdot m^2}{s^3}\] (f) Velocity: \[\frac{m}{s}\] (g) Energy: \[kg\cdot\frac{m^2}{s^2}\]

Step by step solution

01

(a) Acceleration

Acceleration is defined as the rate of change of velocity with time, i.e., distance per time squared. Therefore, the acceleration has SI units of meter for distance (m), which we'll divide by second squared for time (s^2). The derived SI units for acceleration is: \[\frac{m}{s^2}\]
02

(b) Force

Force comes from Newton's second law of motion, which states that force equals mass times acceleration, i.e., F = ma. Mass has the SI unit of kilogram (kg) and acceleration has the derived SI unit of \(\frac{m}{s^2}\). So, the derived SI units for force is: \[kg\cdot\frac{m}{s^2}\]
03

(c) Work

Work is done when a force acts on an object and causes it to move in the direction of the force, i.e., W = Fd. Force has the derived SI unit of \(kg\cdot\frac{m}{s^2}\), and distance has the SI unit of meter (m). Therefore, the derived SI units for work is: \[kg\cdot\frac{m^2}{s^2}\]
04

(d) Pressure

Pressure is defined as force applied per unit area, i.e., P = \(\frac{F}{A}\). Force has derived SI units of \(kg\cdot\frac{m}{s^2}\), and area, being length squared, has the SI units of \(m^2\). Hence, the derived SI units for pressure are: \[\frac{kg}{m\cdot s^2}\]
05

(e) Power

Power is the rate at which work is being done, i.e., P = \(\frac{W}{t}\). Work has the derived SI units of \(kg\cdot\frac{m^2}{s^2}\), and time has the SI unit of second (s). So, the derived SI units for power are: \[\frac{kg\cdot m^2}{s^3}\]
06

(f) Velocity

Velocity is the rate of change of position, i.e., v = \(\frac{d}{t}\). Distance has the SI unit of meter (m), and time has the SI unit of second (s). Therefore, the derived SI units for velocity are: \[\frac{m}{s}\]
07

(g) Energy

Energy can be defined as the kinetic energy of an object, i.e., E = \(\frac{1}{2}mv^2\). Mass has the SI base unit of kilogram (kg), and velocity has the derived SI unit of \(\frac{m}{s}\). So, the derived SI units for energy are: \[kg\cdot\frac{m^2}{s^2}\]

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Key Concepts

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

Acceleration
Acceleration is the term used to describe the rate of change of velocity over time. Imagine you're riding a bicycle and start pedaling faster; you're accelerating. In physics, acceleration is calculated by measuring how quickly your velocity changes. It's expressed in meters per second squared (m/s虏). This is because you measure the increase or decrease of speed per second, for each second it happens. Picking up speed rapidly means a higher acceleration. When you see
  • "meters" for distance,
  • "seconds" for time,
  • put them together as "meters per second squared,"
you get a complete picture of what acceleration means in SI units.
Force
Force makes things move, stop, or change direction. It鈥檚 what makes a car accelerate or a ball thrown into the air come back down. The scientific definition of force involves mass and acceleration. According to Newton鈥檚 second law of motion, force is the product of an object's mass and its acceleration: F = ma. In SI units, you measure mass in kilograms (kg) and acceleration in meters per second squared (m/s虏), so force is measured in newtons (N): 1 N = 1 kg鈰卪/s虏. Think of pushing a shopping cart: The heavier it is, or the faster you want it to move, the more force you need to apply.
  • The formula is F = ma,
  • where "m" is mass and "a" is acceleration.
Work
In physics, work isn't about tasks or chores, but about energy transfer. When you apply a force to an object and it moves in the direction of the force, that's work. The formula is simple: Work (W) = Force (F) 脳 Distance (d). If you push a book across a table, the work you have done depends on how hard you push and how far the book moves. The SI unit is a joule (J), where 1 joule = 1 newton-meter (1 kg鈰卪虏/s虏). To understand it better, picture lifting a weight: You need force to lift it, and the higher you lift, the more work is done. Work can't happen unless there's movement. So, no movement, no work!
Pressure
Pressure tells us how force is distributed over an area. It's the reason why a needle can pierce a piece of cloth while a pen can't, even if the same force is applied. Pressure is calculated as force is divided by area (P = F/A). The unit for pressure in the SI system is the pascal (Pa), equivalent to one newton per square meter (1 N/m虏 or 1 kg/m鈰卻虏). Think of snowshoes; they distribute your weight over a larger area, reducing pressure on the snow so you don't sink. This concept is crucial in fields ranging from meteorology to hydraulics.
Velocity
Velocity describes the speed of something in a given direction. It's a bit more specific than just speed, as it combines how fast something moves with the direction it moves in. This is important because changes in direction also constitute changes in velocity. The formula for it is velocity (v) = distance (d) / time (t). In SI units, you measure it in meters per second (m/s). If a car travels 60 meters in 2 seconds, its velocity is 30 meters per second. Knowing both speed and direction gives a complete picture of an object's motion, which is why velocity is such a critical concept in physics.

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Most popular questions from this chapter

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