Problem 66 - Chemistry: The Central Science

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 \(\times\) distance (d) pressure \(=\) force/area (e) power = work/time (f) velocity \(=\) distance/time (g) energy \(=\) mass \(\times(\text { velocity })^{2}\)

Step-by-Step Solution

Answer

(a) Acceleration derived SI unit: \(a = \frac{m}{s^2}\) (b) Force derived SI unit: \(F = kg \times \frac{m}{s^2}\) (c) Work derived SI unit: \(W = (kg \times \frac{m}{s^2}) \times m\) (d) Pressure derived SI unit: \(P = \frac{kg \times \frac{m}{s^2}}{m^2}\) (e) Power derived SI unit: \(P = \frac{(kg \times \frac{m}{s^2}) \times m}{s}\) (f) Velocity derived SI unit: \(v = \frac{m}{s}\) (g) Energy derived SI unit: \(E = kg \times (\frac{m}{s})^2\)

Step 1: (a) Acceleration derived SI unit

Using the given formula, acceleration (a) = distance (d) / time (t)^2. Thus, the derived SI unit for acceleration is: \(a = \frac{d}{t^2} = \frac{m}{s^2}\)

Step 2: (b) Force derived SI unit

Using the given formula, force (F) = mass (m) × acceleration (a). The derived SI unit for force is: \(F = m \times a = kg \times \frac{m}{s^2}\)

Step 3: (c) Work derived SI unit

Using the given formula, work (W) = force (F) × distance (d). The derived SI unit for work is: \(W = F \times d = (kg \times \frac{m}{s^2}) \times m\)

Step 4: (d) Pressure derived SI unit

Using the given formula, pressure (P) = force (F) / area (A). The derived SI unit for pressure is: \(P = \frac{F}{A} = \frac{kg \times \frac{m}{s^2}}{m^2}\)

Step 5: (e) Power derived SI unit

Using the given formula, power (P) = work (W) /time (t). The derived SI unit for power is: \(P = \frac{W}{t} = \frac{(kg \times \frac{m}{s^2}) \times m}{s}\)

Step 6: (f) Velocity derived SI unit

Using the given formula, velocity (v) = distance (d) /time (t). The derived SI unit for velocity is: \(v = \frac{d}{t} = \frac{m}{s}\)

Step 7: (g) Energy derived SI unit

Using the given formula, energy (E) = mass (m) × (velocity (v))^2. The derived SI unit for energy is: \(E = m \times v^2 = kg \times (\frac{m}{s})^2\)

Key Concepts

Derived Units

Derived units in the International System of Units (SI) are based on the seven base units, like meters and seconds. They form when base units are combined through multiplication or division.
For example, the unit of speed, meters per second (m/s), is derived from the base units of meters and seconds.
Derived units are essential because they enable us to quantify and communicate complex physical phenomena.

• Acceleration: A derived unit, calculated as distance divided by time squared, results in meters per second squared (m/s²).
• Force: Combining mass and acceleration, its unit is derived to be Newton (N), which is equivalent to kg·m/s².
• Energy: Combines mass, distance, and time, and results in joules (J), derived as kg·m²/s².

Derived units allow scientists and engineers to describe how things move, interact, and generate energy.

Acceleration

Acceleration is the rate of change of velocity over time. Think of it as how quickly an object speeds up or slows down.
In physics, it is calculated as the change in velocity divided by the time over which the change occurs. This is the reason why its derived unit is meters per second squared (m/s²).
The formula for acceleration is: \[ a = \frac{\Delta v}{\Delta t} \] Where:

• \(\Delta v\) is the change in velocity (m/s)
• \(\Delta t\) is the time interval (s)

This derived unit helps us understand phenomena like how quickly a car picks up speed or how a spacecraft accelerates in space.

Force

Force is an interaction that changes the motion of an object, like pushing or pulling. It involves the concepts of mass and acceleration.
The formula used to calculate force is: \[ F = m \times a \] Where:

• \(F\) is force in newtons (N)
• \(m\) is mass in kilograms (kg)
• \(a\) is acceleration in meters per second squared (m/s²)

Using this, the derived unit of force, the newton, relates directly to how much an object is being accelerated by a force acting upon it. It's easy to see how the SI unit whole is kg·m/s².

Energy

Energy is the capacity to perform work. It is found in several forms in the physical world, including kinetic, potential, thermal, and more.
In mechanical terms, energy is often associated with movement and the ability to move objects.
The formula for kinetic energy, which is a common form, is: \[ E = \frac{1}{2} m v^2 \] Where:

• \(E\) is energy in joules (J)
• \(m\) is mass in kilograms (kg)
• \(v\) is velocity in meters per second (m/s)

This formula demonstrates how energy can be directly tied to both mass and velocity, helping us understand both everyday occurrences, like driving a car, and complex systems, like electron movement within atoms.