# 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 :