Question

Which of the following relation is not dimensionally correct? (K→kinetic energy, m→mass, v→speed, h→height, g→acceleration due to gravity) *
a)K = m²v³
b) K = mv²
c) K = (3/16)mv²
d) K = mgh​

Answer 1

Which of the following relation is not dimensionally correct ?

a) K = m²v³

b) K = mv²

c) K = (3/16) mv²

d) K = mgh

solution : let's check all the options.

a) K = m²v³

here K is kinetic energy so dimensions of K = [ML²T¯²]

so, LHS = [ML²T¯²]

RHS = m²v³ = [M²][L³T¯³] = [M²L³T¯³]

we see, LHS ≠ RHS

so, option (a) is not dimensionally correct.

b) K = mv²

LHS = [ML²T¯²]

RHS = mv² = [M][LT¯¹]² = [ML²T¯²]

Here, LHS = RHS

so option (b) is dimensionally correct.

c) K = (3/16) mv²

here , (3/16) is a constant and dimensionless quantity.

so, LHS = dimensions of K = [ML²T¯²]

RHS = dimensions of (3/16) mv² = dimensions of mv² = [ML²T¯²]

here, LHS = RHS

hence, option (c) is also dimensionally correct.

d) K = mgh

LHS = [ML²T¯²]

RHS = mgh = [M][LT¯²][L] = [ML²T¯²]

here, LHS = RHS

so, option (d) is dimensionally correct.

Therefore the option (a) is the only relation which is not dimensionally correct.