Problem 72 - Chemistry The Central Science In Si Units

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) m/s², (b) N, (c) J, (d) Pa, (e) W, (f) m/s, (g) J.

Step 1: Analyzing Acceleration

Acceleration is given by the formula \( a = \frac{d}{t^2} \), where \( d \) is distance and \( t \) is time. In base SI units, distance is measured in meters (m) and time is in seconds (s). Thus, the derived SI unit for acceleration is \( \frac{\text{m}}{\text{s}^2} \), or \( \text{m/s}^2 \).

Step 2: Determining Force Units

Force is calculated using \( F = m \cdot a \), where \( m \) is mass and \( a \) is acceleration. Mass in SI units is in kilograms (kg) and acceleration is in meters per second squared (m/s²). Therefore, the derived unit for force is \( \text{kg}\cdot\text{m/s}^2 \), known as a newton (N).

Step 3: Calculating Work Units

Work is given by \( W = F \cdot d \), where \( F \) is force and \( d \) is distance. The derived unit for force is newtons (N) and distance is in meters (m). Thus, work is measured in newton-meters (N⋅m), also known as a joule (J).

Step 4: Establishing Pressure Units

Pressure is defined as \( P = \frac{F}{A} \), where \( F \) is force and \( A \) is area. The force is measured in newtons (N) and area is in square meters (m²). Thus, the derived SI unit for pressure is \( \frac{\text{N}}{\text{m}^2} \), which is called a pascal (Pa).

Step 5: Deriving Power Units

Power is defined by \( P = \frac{W}{t} \), where \( W \) is work and \( t \) is time. Work is measured in joules (J) and time in seconds (s). Therefore, the unit for power is \( \frac{\text{J}}{\text{s}} \), known as a watt (W).

Step 6: Analyzing Velocity Units

Velocity is given by \( v = \frac{d}{t} \), with distance \( d \) in meters (m) and time \( t \) in seconds (s). Thus, the unit for velocity is meters per second (m/s).

Step 7: Calculating Energy Units

Energy is calculated using \( E = m \cdot v^2 \). Mass \( m \) is in kilograms (kg) and velocity \( v \) is in meters per second (m/s), squared in this case. Thus, the derived unit for energy is \( \text{kg}\cdot(\text{m/s})^2 \), which simplifies to a joule (J).

Key Concepts

Acceleration

Acceleration is a measure of how quickly something speeds up or slows down. It involves a change in velocity over time. To calculate acceleration, we use the formula \( a = \frac{d}{t^2} \), where \( d \) is distance and \( t \) is time.

• Distance is measured in meters \( (\text{m}) \).
• Time is measured in seconds \( (\text{s}) \).

Therefore, the unit of acceleration is meters per second squared, expressed as \( \text{m/s}^2 \). This tells us how much the speed changes every second.

Force

Force describes the push or pull on an object, affecting its motion. It is calculated using the formula \( F = m \cdot a \), where:

• \( m \) is mass in kilograms \( (\text{kg}) \).
• \( a \) is acceleration in meters per second squared \( (\text{m/s}^2) \).

The resulting unit for force is \( \text{kg} \cdot \text{m/s}^2 \), known as a newton \( (\text{N}) \). One newton is the amount of force required to accelerate a one-kilogram mass by one meter per second squared.

Work

Work refers to the energy transferred when a force moves an object over a distance. The equation for work is \( W = F \cdot d \), where:

• \( F \) is force in newtons \( (\text{N}) \).
• \( d \) is distance in meters \( (\text{m}) \).

The derived unit for work is the newton-meter, or joule \( (\text{J}) \). This unit indicates how much energy is used to move an object.

Pressure

Pressure is the amount of force applied per unit area. It is defined by the formula \( P = \frac{F}{A} \), where:

• \( F \) is force in newtons \( (\text{N}) \).
• \( A \) is area in square meters \( (\text{m}^2) \).

The resulting unit of pressure is newtons per square meter, known as a pascal \( (\text{Pa}) \). This tells us how force is spread over an area.

Power

Power measures how quickly work is done or energy is transferred. It is calculated using \( P = \frac{W}{t} \), which breaks down as follows:

• \( W \) is work in joules \( (\text{J}) \).
• \( t \) is time in seconds \( (\text{s}) \).

The power unit is joules per second, more commonly known as watts \( (\text{W}) \). A watt represents one joule of work done in one second.

Velocity

Velocity describes the speed of an object in a particular direction. It is expressed by \( v = \frac{d}{t} \), where:

• \( d \) represents distance in meters \( (\text{m}) \).
• \( t \) is time in seconds \( (\text{s}) \).

The unit for velocity is meters per second \( (\text{m/s}) \). This indicates both how fast and in which direction an object is moving.

Energy

Energy is the capacity to do work. It is determined with \( E = m \cdot v^2 \), involving:

• \( m \) as mass in kilograms \( (\text{kg}) \).
• \( v \) as velocity in meters per second \( (\text{m/s}) \).

Squaring velocity means multiplying it by itself, \( (\text{m/s})^2 \). The derived SI unit for energy is the joule \( (\text{J}) \), indicating the ability of an object to perform work.