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प्रश्न
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 `(2pit)/T`
(b) y = a sin vt
(c) y = `(a/T) sin t/a`
d) y = `(a/sqrt2) (sin 2πt / T + cos 2πt / T )`
(a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds.
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उत्तर १
a) Correct
y = `asin (2pit)/T`
Dimension of y = M0 L1 T0
Dimension of a = M0 L1 T0
Dimension of `sin (2pit)/T = M^0L^0T^0`
∵ Dimension of L.H.S = Dimension of R.H.S
Hence, the given formula is dimensionally correct.
b) Incorrect
Dimension of y = M0 L1 T0
Dimension of a = M0 L1 T0
Dimension of vt = M0 L1 T–1 × M0 L0 T1 = M0 L1 T0
But the argument of the trigonometric function must be dimensionless, which is not so in the given case. Hence, the given formula is dimensionally incorrect.
3) Incorrect
`y =(a/T) sin(t/a)`
Dimension of y = M0L1T0
Dimension of `a/T` = `M^0L^1T^(-1)`
Dimension of `t/a` = `M^0L^(-1)T^1`
But the argument of the trigonometric function must be dimensionless, which is not so in the given case. Hence, the formula is dimensionally incorrect.
4)Correct
y = `(asqrt2)(sin2pi t/T + cos2pi t/T)`
Dimension of y = M0 L1 T0
Dimension of a = M0 L1 T0
Dimension of `t/T` = M0 L0 T0
Since the argument of the trigonometric function must be dimensionless (which is true in the given case), the dimensions of y and a are the same. Hence, the given formula is dimensionally correct.
उत्तर २
According to dimensional analysis an equation must be dimensionally homogeneous.
a) `y = a sin (2pit)/T`
Here, [L.H.S] = [y] = [L] and [R.H.S] = `[a sin (2pit)/T]` = `[L sin T/T]` = [L]
So, it is correct.
b) y = a sin vt
Here [y] = [L] and `[a sin vt] = [L sin(LT^(-1) T)] = [L sinL`]
So the equation is wrong
c) `y = (a/T) sin t/a`
Here [y] = [L] and `[(a/T) sin t/a] = [L/T sin T/L] = [LT^(-1) sin TL^(-1)]`
So the equation is wrong.
d) y = `(asqrt2)(sin (2pit)/T + cos (2pit)/T)`
Here, [y] = [L],[`asqrt2`] = [L]
and` [sin (2pit)/T + cos (2pit)/T] = [sin T/T + cos T/T]` = dimensionless
So the equation is correct
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