1.1.3 Derived Units

Derived units are units that are derived from the seven fundamental SI units (mass, length, time, etc.). In mechanics, derived units are crucial as they describe various physical quantities such as force, energy, power, and more. Understanding these units helps in solving problems related to motion, forces, and energy.

Common Derived Units in Mechanics

1. Velocity

• Unit: Metres per second $(m/s)$
• Derived From: Length (metres) and Time (seconds)
• Formula: $\text{velocity} = \frac{\text{displacement}}{\text{time}}$
• Description: Velocity is the rate at which an object changes its position. It's a vector quantity, meaning it has both magnitude and direction.

2. Acceleration

• Unit: Metres per second squared $(m/s²)$
• Derived From: Velocity $(m/s)$ and Time (seconds)
• Formula: $\text{acceleration} = \frac{\text{change in velocity}}{\text{time}}$
• Description: Acceleration is the rate at which an object's velocity changes with time. It's also a vector quantity.

3. Force

• Unit: Newton $(N)$
• Derived From: Mass (kilograms), Length (metres), and Time (seconds)
• Formula: $\text{force} = \text{mass} \times \text{acceleration}$ or $F = ma$
• Description: Force is the interaction that causes an object to change its velocity. One Newton is the force required to accelerate a $1$ kg mass by $1$ $m/s².$

4. Momentum

• Unit: Kilogram metres per second $(kg·m/s)$
• Derived From: Mass (kilograms) and Velocity (metres per second)
• Formula: $\text{momentum} = \text{mass} \times \text{velocity}$ or $p = mv$
• Description: Momentum is the product of an object's mass and its velocity. It is a vector quantity, indicating both the amount of motion and the direction.

5. Energy/Work

• Unit: Joule $(J)$
• Derived From: Force (Newtons) and Length (metres)
• Formula: $\text{work} = \text{force} \times \text{distance}$ or $W = Fd$
• Description: Energy is the capacity to do work, and work is done when a force moves an object over a distance. One Joule is the work done when a force of $1 N$moves an object $1$ metre.

6. Power

• Unit: Watt $(W)$
• Derived From: Energy (Joules) and Time (seconds)
• Formula: $\text{power} = \frac{\text{work}}{\text{time}}$ or $P = \frac{W}{t}$
• Description: Power is the rate at which work is done or energy is transferred. One Watt is equivalent to one Joule per second.

7. Pressure

• Unit: Pascal $(Pa)$
• Derived From: $Force$ (Newtons) and $Area$ (square metres)
• Formula: $\text{pressure} = \frac{\text{force}}{\text{area}}$ or $P = \frac{F}{A}$
• Description: Pressure is the force exerted per unit area. One Pascal is the pressure exerted by a force of $1$ N acting on an area of $1$ m².

8. Density

• Unit: Kilograms per cubic metre ($kg/m³$)
• Derived From: Mass (kilograms) and Volume (cubic metres)
• Formula: $\text{density} = \frac{\text{mass}}{\text{volume}}$ or $\rho = \frac{m}{V}$
• Description: Density is the mass per unit volume of a substance.

9. Angular Velocity

• Unit: Radians per second ($rad/s)
• Derived From: $Angle$ (radians) and $Time$ (seconds)
• Formula: $\text{angular velocity} = \frac{\text{angle}}{\text{time}}$
• Description: Angular velocity is the rate of change of angular displacement with time.

Using Derived Units