###### Advertisements

###### Advertisements

A famous relation in physics relates ‘moving mass’ *m *to the ‘rest mass’ *m*_{0} of a particle in terms of its speed *v *and the speed of light, *c*. (This relation first arose as a consequence of special relativity due to Albert Einstein). A boy recalls the relation almost correctly but forgets where to put the constant c. He writes:

`m = m_0/(1-v^2)^(1/2)`

Guess where to put the missing c.

###### Advertisements

#### Solution 1

Given the relation,

m = `m_0/(1-v^2)^(1/2)`

Dimension of *m* = M^{1} L^{0} T^{0}

Dimension of `m_0` = M^{1} L^{0} T^{0}

Dimension of *v* = M^{0} L^{1} T^{–1}

Dimension of *v*^{2} = M^{0} L^{2} T^{–2}

Dimension of *c* = M^{0} L^{1} T^{–1}

The given formula will be dimensionally correct only when the dimension of L.H.S is the same as that of R.H.S. This is only possible when the factor, `(1-v^2)^(1/2)` is dimensionless i.e., (1 – *v*^{2}) is dimensionless. This is only possible if *v*^{2} is divided by c^{2}. Hence, the correct relation is `m = m_0/(1-v^2/c^2)^(1/2)`

#### Solution 2

From the given equation, m_0/m = `sqrt(1-v^2)`

Left hand side is dimensionless

Therefore, right hand side should also be dimensionless.

It is possible only when `sqrt(1-v^2) should be sqrt(1-v^2/c^2)

Thus the correct formula is m = `m_0 (1 - v^2/c^2)^((-1)/2)`

#### APPEARS IN

#### RELATED QUESTIONS

Fill in the blanks

The volume of a cube of side 1 cm is equal to.....m^{3}

A new unit of length is chosen such that the speed of light in vacuum is unity. What is the distance between the Sun and the Earth in terms of the new unit if light takes 8 min and 20 s to cover this distance?

Fill in the blanks by suitable conversion of units: 1 kg m^{2}s^{–2}= ....g cm^{2} s^{–2}

Fill in the blanks by suitable conversion of units:

G= 6.67 × 10^{–11} N m^{2} (kg)^{–2}=.... (cm)^{3}s^{–2} g^{–1}.

In which of the following systems can scientific data be exchanged between different parts of the world?

Young’s modulus of steel is 1.9 × 10^{11} N/m^{2}. When expressed in CGS units of dynes/cm^{2}, it will be equal to (1N = 10^{5} dyne, 1m^{2} = 10^{4} cm^{2})

Photon is quantum of radiation with energy E = h ν where ν is frequency and h is Planck’s constant. The dimensions of h are the same as that of ______.

- Linear impulse
- Angular impulse
- Linear momentum
- Angular momentum

During a total solar eclipse the moon almost entirely covers the sphere of the sun. Write the relation between the distances and sizes of the sun and moon.

If the unit of force is 100 N, unit of length is 10 m and unit of time is 100 s, what is the unit of mass in this system of units?

Give an example of a constant which has no unit.

A new system of units is proposed in which unit of mass is α kg, unit of length β m and unit of time γ s. How much will 5 J measure in this new system?