Question

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 \(x\) distance; (d) pressure \(=\) force/area; (e) power = work/time.

Short answer

The derived SI units for each quantity in base SI units are as follows: (a) Acceleration: \(\frac{\text{m}}{\text{s}^2}\) (b) Force: \(\text{kg} \cdot \text{m} \cdot \text{s}^{-2}\) (c) Work: \(\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}\) (d) Pressure: \(\text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-2}\) (e) Power: \(\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-3}\)

Step-by-step solution

(a) Acceleration = distance/time²

First, we need to know the base SI units for distance and time. The base unit for distance is the meter (m), while the base unit for time is the second (s). Therefore, the derived SI unit for acceleration (distance/time²) is given by: \[ \text{Acceleration} ~=~ \frac{\text{Distance}}{\text{Time}^2} \] \[ \text{Acceleration} ~=~ \frac{\text{m}}{\text{s}^2} \]

(b) Force = mass × acceleration

To find the derived SI unit for force, we need to know the base SI unit for mass, which is the kilogram (kg). We know from the previous step that the derived SI unit for acceleration is m/s². So, to find the derived SI unit for force (mass × acceleration), we have: \[ \text{Force} ~=~ \text{Mass} \times \text{Acceleration} \] \[ \text{Force} ~=~ \text{kg} \times \frac{\text{m}}{\text{s}^2} = \text{kg} \cdot \text{m} \cdot \text{s}^{-2} \]

(c) Work = force × distance

To find the derived SI unit for work, we will use the derived SI unit for force that we found in the previous step (kg·m·s⁻²) and the base SI unit for distance (m). Therefore, the derived SI unit for work (force × distance) is given by: \[ \text{Work} ~=~ \text{Force} \times \text{Distance} \] \[ \text{Work} ~=~ \text{kg} \cdot \text{m} \cdot \text{s}^{-2} \times \text{m} = \text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2} \]

(d) Pressure = force/area

We already found the derived SI unit for force (kg·m·s⁻²) in step (b). To find the derived SI unit for pressure, we need to know the base SI unit for the area, which is m². Therefore, the derived SI unit for pressure (force/area) is given by: \[ \text{Pressure} ~=~ \frac{\text{Force}}{\text{Area}} \] \[ \text{Pressure} ~=~ \frac{\text{kg} \cdot \text{m} \cdot \text{s}^{-2}}{\text{m}^2} = \text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-2} \]

(e) Power = work/time

From step (c), we found the derived SI unit for work to be kg·m²·s⁻². We already know the base SI unit for time, which is s. Now, to find the derived SI unit for power (work/time), we have: \[ \text{Power}~=~ \frac{\text{Work}}{\text{Time}} \] \[ \text{Power} ~=~ \frac{\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}}{\text{s}} = \text{kg} \cdot \text{m}^2 \cdot \text{s}^{-3} \]

Key concepts

Understanding Acceleration

Acceleration is all about how quickly an object speeds up or slows down. It's a measure of change in velocity over time. To determine acceleration in terms of SI units, we start with the basics: distance and time. The base SI unit for distance is the meter (m), and for time, it's the second (s). Acceleration is defined as the change in velocity per unit time, or essentially, distance covered per second squared.

• This means the derived SI unit for acceleration is meters per second squared (\(\text{m/s}^2\)).
• It captures the notion of speed change occurring in each second.

Understanding acceleration in this way helps us grasp more complex concepts like force and work.

Exploring Force

Force provides a push or pull that affects the motion of an object. In the SI unit system, force is derived from the relationship between mass and acceleration. Mass has the base unit of kilogram (kg), while we previously identified acceleration as having the unit \(\text{m/s}^2\). To find the derived unit of force, multiply these two components:

• Force = Mass \(\times\) Acceleration
• The derived SI unit would thus be kilogram meter per second squared (\(\text{kg} \cdot \text{m/s}^2\)).

This derived unit has its own special name—the Newton (N)—in honor of Sir Isaac Newton. It's helpful to think of force as the heart of motion in physics.

Understanding Work

Work is a measure of energy transfer when a force moves an object over a distance. The concept of work links force to the distance it moves something. In the realm of SI units, we utilize our previously calculated force unit.

• Work is computed as Force \(\times\) Distance.
• Given our force is in Newtons (\(\text{kg} \cdot \text{m/s}^2\)), and distance in meters (m), the unit becomes kilogram meter squared per second squared (\(\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}\)).

This unit is also known as the Joule (J). Whenever work is done, energy is either consumed or produced. Understanding this helps when studying how machines and humans overcome forces on a daily basis.

Unpacking Pressure

Pressure involves how much force is applied over a certain area, indicating how concentrated that force is on a surface. With pressure, the SI unit components rearrange slightly from those used in force calculations.

• Pressure = Force / Area
• Our force unit from before is the Newton (N), which is \(\text{kg} \cdot \text{m/s}^2\), and area has the unit square meters (m²).

Thus, the derived SI unit for pressure is kilogram per meter per second squared (\(\text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-2}\)). This unit is also called the Pascal (Pa). Pressure tells us how distributed the exerted force is over an area, a critical concept in fields like aerodynamics, meteorology, and medicine.

Power Demystified

Power is about how fast work is done or energy is transferred. It combines concepts of work and time, giving insight into the rate at which energy is used.

• Power = Work / Time
• Since work is measured in Joules (\(\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}\)), and time in seconds (s), the unit for power becomes kilogram meter squared per second cubed (\(\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-3}\)).

This unit has its unique name: the Watt (W), named after James Watt. Recognizing power's importance helps when we consider how quickly energy sources or machines perform work, such as in engines or electrical devices.