Physical quantities Units Dimensions – Problems and Solutions

1.

Based on the above table, which quantities have true units and dimensions.

Solution :

1) Momentum

The equation of momentum is p = m v

*p = momentum, m = mass, v = velocity*

Dimension of mass = M and dimension of velocity = L/T = L T^{-1 }so that dimension of momentum = **M L T**^{-1}

International unit of momentum = kg m/s = **kg m s**^{-1}

2) Force

The equation of force is F = m a

*F = force, m = mass**, a = acceleration*

Dimension of mass = M and dimension of acceleration = L/T^{2} = L T^{-2} so dimension of force is **M L T**^{-2}

International unit of force is kg m/s^{2} = **kg m s**^{-2}

3) Power

The equation of power is W = F d

*W = work, F = force, d = displacement*

Dimension of force = M L T^{-2} and dimension of displacement is L so that dimension of work is [M][L][T]^{-2 }[L] = [M][L]^{2}[T]^{-2}

The equation of power is P = W / t

*P = power, W = work, t = time*

Dimension of work = [M][L]^{2}[T]^{-2 }and dimension of time = [T] so that dimension of power = [M][L]^{2}[T]^{-2 }/ [T] = [M][L]^{2}[T]^{-2 }[T]^{-1 } = [M][L]^{2}[T]^{-3}

International unit of force is kg m^{2}/s^{3 }= kg m^{2 }s^{-3}

2. Based on table below, quantities with correct units and dimension are….

Solution :

The equation of momentum is **p = m v**.

Unit of mass (m) is kilogram (kg) and unit of velocity (v) is meter per second (m/s) so that unit of momentum is **kg m/s or kg m/s**. Kilogram is the dimension of mass with dimension of [M], meter is a unit of length with a dimension of [L], second is the unit of time with dimension of [T] so that dimension of momentum is **[M][L]/[T] or [M][L][T]**^{-1}**. **

The equation of force is **F = m a**.

Unit of Mass (m) is kilogram (kg) and unit of acceleration (a) is meters per second squared (m/s^{2}) so the unit of force is **kg m/s**^{2 }**or kg m s**^{-2}. Unit of mass is kilogram with dimension of [M], unit of length is meter with dimension of [L], unit of time is second with dimension of [T] so that dimension of force is **[M][L]/[T]**^{2 }**or [M][L][T]**^{-2}

The equation of power is **P = W/t**, the equation of work is W = F s, the equation of force is F = m a.

Unit of mass is kilogram (kg), unit of acceleration is meters per second squared (m/s^{2}) so that unit of force is kg m/s^{2}. Unit of displacement is meter (m), unit of force is kg m/s^{2} so that unit of work is kg m/s^{2 }x m = kg m^{2}/s^{2}. Unit of time is second (s), a unit of work is kg m^{2}/s^{2 }so that unit of power is kg m^{2}/s^{2 }: s = **kg m**^{2}**/s**^{3}** or kg m**^{2}** s**^{-3}.

Unit of mass is kilogram with dimension of [M], unit of length is meter with dimension of [L], unit of time is second with the dimension of [T] so that dimension of power is **[M][L]**^{2}**/[T]**^{3 }**or [M][L]**^{2}**[T]**^{-3}**.**

3. Power is defined as the rate at which work is done. Or power is the ratio of work to the time interval. Determine the dimension of power.

Solution :

The equation of power :

*W = work, F = power, a = acceleration, v = velocity, d = distance, t = time interval*

*m = mass (dimension of mass = M), d = distance (dimension of distance = L), t = time (dimension of time = T).*

Dimension of power :

**What is a physical quantity?****Answer:**A physical quantity is a property of an object or system that can be quantified and measured. Examples include mass, length, time, temperature, and force.

**Why are units important in the measurement of physical quantities?****Answer:**Units provide a standard way to express the magnitude of a physical quantity. They ensure clarity, precision, and consistency in measurements, enabling clear communication and understanding among scientists and engineers worldwide.

**What is the difference between a fundamental (or base) quantity and a derived quantity?****Answer:**Fundamental quantities are basic physical quantities defined independently and form the foundation for other measurements. Examples include length, mass, and time. Derived quantities are formed from combinations of these fundamental quantities, such as area (length x width) or velocity (distance/time).

**What are dimensions? How do they relate to physical quantities?****Answer:**Dimensions refer to the nature and type of physical quantities (e.g., [L] for length, [M] for mass, [T] for time). They represent the powers to which the fundamental quantities are raised to represent a particular physical quantity.

**What is meant by the “dimensional formula” of a physical quantity?****Answer:**A dimensional formula expresses the relationship of a physical quantity in terms of its basic dimensions. For example, the dimensional formula for velocity is $[LT]$, indicating that velocity is derived from length ([L]) divided by time ([T]).

**How is the SI (International System of Units) system significant in modern science and engineering?****Answer:**The SI system provides a globally accepted set of standard units for measuring physical quantities. This ensures uniformity, reduces confusion, and enables collaboration among scientists and engineers worldwide.

**What’s the difference between a scalar and a vector quantity?****Answer:**A scalar quantity has only magnitude (e.g., mass, temperature), whereas a vector quantity has both magnitude and direction (e.g., velocity, force).

**Why can’t we use arbitrary units to measure physical quantities?****Answer:**Using arbitrary units would lead to confusion, lack of consistency, and miscommunication in scientific measurements and calculations. Standardized units ensure that measurements are universally understood and comparable.

**How can dimensional analysis be useful in physics?****Answer:**Dimensional analysis helps verify the correctness of physical equations by checking the consistency of dimensions on both sides of the equation. It can also be used to derive relationships between different physical quantities.

**Why are some quantities, like refractive index or coefficient of friction, considered dimensionless?**

**Answer:**Dimensionless quantities are ratios of similar quantities, and thus their dimensions cancel out. For instance, the refractive index is the ratio of the speed of light in a vacuum to its speed in a medium. Since both are speeds, their dimensions cancel, making the refractive index dimensionless.