A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion : (a) y = a sin 2π t/T (b) y = a sin υt (c) y = (a/T) sin t/a (d) y = (a √2) (sin 2πt / T + cos 2πt / T ) (a = maximum displacement of the particle, υ = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds.
(a)
The displacement of a particle undergoing a certain periodic motion is given as,
y = a sin 2 π t / T
The dimension of displacement y and a is [L]. The dimension of time t and T is [T].
Write the equation of the displacement in term of dimension.
[ L ] = [ L ] [ sin2 π [T] [T] ] [ L ] = [ L ]
Here the dimension on LHS and RHS are same so the given equation is dimensionally correct.
(b)
The displacement of a particle undergoing a certain periodic motion is given as,
y = a sin v t
The dimension of displacement y and a is [L]. The dimension of time t and T is [T]. The dimension of velocity v is [ LT − 1 ] .
Write the equation of the displacement in term of dimension.
[ L ] = [ L ] [ sin[LT − 1 ][T] ] [ L ] = [ L ] [ sin[L] ] [ L ] ≠ [ L ]
The argument of a trigonometry function must be dimensionless. But in this case the argument of a trigonometry function is not dimensionless. So the equation is dimensionally incorrect.
(c)
The displacement of a particle undergoing a certain periodic motion is given as,
y = ( a / T ) sin t / a
The dimension of displacement y and a is [L]. The dimension of time t and T is [T].
Write the equation of the displacement in term of dimension.
[ L ] = [ L ] [ T ] [ sin [ T ] [ L ] ] [ L ] ≠ [ L ] [ T ] [ sin [ T ] [ L ] ]
The argument of a trigonometry function must be dimensionless. But in this case the argument of a trigonometry function is not dimensionless. So the equation is dimensionally incorrect.
(d)
The displacement of a particle undergoing a certain periodic motion is given as,
y = ( a 2 ) ( sin 2 π t / T + cos 2 π t / T )
The dimension of displacement y and a is [L]. The dimension of time t and T is [T].
Write the equation of the displacement in term of dimension.
[ L ] = [ L ] [ sin [ T ] [ T ] + cos [ T ] [ T ] ] [ L ] = [ L ]
Here the dimension on LHS and RHS are same so the given equation is dimensionally correct.
(a)
The displacement of a particle undergoing a certain periodic motion is given as,
The dimension of displacement y and a is [L]. The dimension of time t and T is [T].
Write the equation of the displacement in term of dimension.
Here the dimension on LHS and RHS are same so the given equation is dimensionally correct.
(b)
The displacement of a particle undergoing a certain periodic motion is given as,
The dimension of displacement y and a is [L]. The dimension of time t and T is [T]. The dimension of velocity
Write the equation of the displacement in term of dimension.
The argument of a trigonometry function must be dimensionless. But in this case the argument of a trigonometry function is not dimensionless. So the equation is dimensionally incorrect.
(c)
The displacement of a particle undergoing a certain periodic motion is given as,
The dimension of displacement y and a is [L]. The dimension of time t and T is [T].
Write the equation of the displacement in term of dimension.
The argument of a trigonometry function must be dimensionless. But in this case the argument of a trigonometry function is not dimensionless. So the equation is dimensionally incorrect.
(d)
The displacement of a particle undergoing a certain periodic motion is given as,
The dimension of displacement y and a is [L]. The dimension of time t and T is [T].
Write the equation of the displacement in term of dimension.
Here the dimension on LHS and RHS are same so the given equation is dimensionally correct.
