Problem 72 - Chemistry: The Central Science in SI Units, Global Edition

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: \(\frac{m}{s^2}\) (b) Force: \(kg*\frac{m}{s^2} \) or N (Newton) (c) Work: \(kg*\frac{m^2}{s^2} \) or J (Joule) (d) Pressure: \(\frac{kg}{m*s^2} \) or Pa (Pascal) (e) Power: \(\frac{kg*m^2}{s^3} \) or W (Watt) (f) Velocity: \(\frac{m}{s}\) (g) Energy: \(kg*\frac{m^2}{s^2}\) or J (Joule)

Step 1: (a) SI units of Acceleration

Acceleration is defined as the rate of change of velocity per unit of time, so its unit is distance(divided by)time^(2). The SI base unit of distance is meter(m), and that of time is second(s). So, the SI units of acceleration is m/s^2.

Step 2: (b) SI units of Force

Force is defined as mass times acceleration. So, its unit is mass multiplied by acceleration. We know from above, that the SI units of acceleration is m/s^2, and mass is kg. So, the SI units of force is kg*m/s^2, which is also known as a Newton (N) in honor of Sir Isaac Newton.

Step 3: (c) SI units of Work

Work is defined as force times distance. Therefore, its unit is force multiplied by distance. From above, we know that the SI units of force is kg*m/s^2, and distance is m. So, the SI units of work is kg*m^2/s^2, which is also known as Joule (J).

Step 4: (d) SI units of Pressure

Pressure is defined as force per unit area. Therefore, the unit of pressure is force divided by area. We know from above, the SI units of force is kg*m/s^2, and area is m^2. So, the SI units of pressure is kg/m*s^2, which is also known as Pascal (Pa).

Step 5: (e) SI units of Power

Power is defined as work per time. So, its unit is work divided by time. From above, we know that the SI units of work is kg*m^2/s^2, and time is s. So, the SI units of power is kg*m^2/s^3, which is also known as Watt (W).

Step 6: (f) SI units of Velocity

Velocity is defined as distance covered per unit time, or displacement per time. Therefore, its unit is distance divided by time. Using the SI base units for distance (meters, m) and time (seconds, s), the SI units of velocity is m/s.

Step 7: (g) SI units of Energy

Energy is defined as mass times the square of velocity. Therefore, its unit is mass multiplied by velocity squared. We know from above, that the SI units of mass is kg and the velocity is m/s. So, the SI units of energy is kg*m^2/s^2, which is also known as Joule (J).

Key Concepts

Acceleration

Acceleration is a key concept that measures how quickly an object's velocity changes over time. It is often referred to when observing objects in motion and can determine whether an object is speeding up, slowing down, or changing direction. The formula for acceleration is the change in velocity divided by the change in time. In SI units, velocity is measured in meters per second (m/s), and time is measured in seconds (s).

Thus, the unit of acceleration becomes meters per second squared (\( \text{m/s}^2 \)). The term "per second squared" essentially means that velocity changes by a certain number of meters per second, each second. When you apply acceleration in real-world scenarios, you often describe this as how quickly something is getting faster or slower. For example, a car accelerating from a stop to a speed of 20 m/s in 10 seconds has an acceleration of 2 m/s².

Understanding the unit \( \text{m/s}^2 \) can help grasp how forces influence motion, which brings us to our next topic.

Force

Force is an interaction that, when unopposed, changes the motion of an object. According to Newton's second law of motion, it is the action required to accelerate an object with a given mass. In simple terms, force can be thought of as a push or pull on an object. The formula to calculate force is \( \text{Force (F) = mass (m) \times acceleration (a)} \).

In SI units, mass is measured in kilograms (kg), and acceleration is measured in meters per second squared (\( \text{m/s}^2 \)). Therefore, the SI unit of force is a newton (N), defined as the force required to accelerate a one-kilogram mass by one meter per second squared. This is expressed as \( \text{N = kg} \cdot \text{m/s}^2 \).

For instance, consider pushing a stationary block lying on a frictionless surface. To make this block accelerate, you need to apply a force. This application of force and its resultant effect is fundamental to understanding mechanics and how objects interact with each other.

Energy

Energy is the capacity to do work, and it is an essential quantity in physics and everyday life. It appears in various forms, such as kinetic energy, potential energy, and thermal energy, among others. In mechanics, energy is often derived when a force causes an object to move. One of the fundamental types of energy is kinetic energy, which is related to an object's mass and velocity.

The equation linking energy to these properties is \( \text{Energy (E) = mass (m) \times (velocity (v))^2} \). In SI units, mass is measured in kilograms (kg) and velocity in meters per second (m/s). As a result, the SI unit of energy is the joule (J), where one joule is equal to one kilogram meter squared per second squared (\( \text{J = kg} \cdot \text{m}^2/\text{s}^2 \)).

This concept helps explain how energy can be transformed and conserved, influencing not only physics but also disciplines like engineering and biology. For example, in a moving car, energy is harnessed from fuels and transformed into kinetic energy, propelling the vehicle forward. Understanding energy and its calculations allows us to efficiently harness and utilize power in various technologies and natural processes.