Question

Which of the following relations is dimensionally incorrect? *
(a) ½ mv² = mgh
(b) K = m²v³
(c) W = ½ mv² – ½ mu²
(d) T = √(l/g)​

Answer 1

A> We know acceleration a=rate of change of velocity
a=
t
v−u
​
where v=final velocity and u = initial velocity and t= time
Thus we can rewrite the above equation as,
u=v−at
[LT
−1
]=[LT
−1
]−[LT
−2
][T]
[LT
−1
]=[LT
−1
] (Dimensionally correct)

B> From equation of motion we get displacement
s=ut+
2
1
​
at
2

s−ut=
2
1
​
at
2

[L]−[LT
−1
][T]=[LT
−2
][T
2
]
Here also, LHS =RHS

C>The equation u
2
=2a(gt−1)
Substituting the dimensions:
[LT
−1
]
2
=[LT
−2
]([LT
−2
][T])
[L
2
T
−2
]=[L
2
T
−3
]
LHS is not equal to RHS.

D> From equation of motion,
v
2
=u
2
+2as
v
2
−u
2
=2as
[LT
−1
]
2
−[LT
−1
]
2
=[LT
−2
][L]
So, LHS=RHS

Whereas the dimensions of the equation are not the same on the LHS and RHS.
Thus option C is incorrect

Answer 2

Answer:

b) is wrong

As, K means force constant have a dimension MT^-2 but m^2 v^3 have dimension of M^2L^2T^-2.

hope this will help you