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(ncert): 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 Vt (c) Y = (a/t) Sin T/a (d) Y = (a 2) (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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Answer»
Dimension of a = displacement = [M0L1T0] Dimension of V (speed) = distance/time = [M0L1T-1] Dimension of t or T (time period) = [M0L0T1] Trigonometric function sine is a ratio, hence it must be dimensionless. (a) y = a sin 2π t/T (correct ✓ ) Dimensions of RHS = [L1] sin([T].[T-1] ) = [M0L1T0] = LHS (eqation is correct). (b) y = a sin vt (wrong ✗) RHS = [L1] sin([LT-1] [T1]) = [L1] sin([L]) = wrong, since trigonometric function must be dimension less. (c) y = (a/T) sin t/a (wrong ✗) RHS = [L1] sin([T].[L-1] ) = [L1] sin([TL-1] ) = wrong, sine function must be dimensionless. (d) y = (a 2) (sin 2πt / T + cos 2πt / T ) (correct ✓ ) RHS = [L1] ( sin([T].[T-1] + cos([T].[T-1] ) = [L1] ( sin(M0L1T0) + cos(M0L1T0) ) = [L1] = RHS = equation is dimensionally correct. Given, Dimension of a = displacement = [M0L1T0] Dimension of v (speed) = distance/time = [M0L1T-1] Dimension of t or T (time period) = [M0L0T1] Trigonometric function sine is a ratio, hence it must be dimensionless. (a) y = a sin 2π t/T (correct ✓ ) Dimensions of RHS = [L1] sin([T].[T-1] ) = [M0L1T0] = LHS (eqation is correct). (b) y = a sin vt (wrong ✗) RHS = [L1] sin([LT-1] [T1]) = [L1] sin([L]) = wrong, since trigonometric function must be dimension less. (c) y = (a/T) sin t/a (wrong ✗) RHS = [L1] sin([T].[L-1] ) = [L1] sin([TL-1] ) = wrong, sine function must be dimensionless. (d) y = (a 2) (sin 2πt / T + cos 2πt / T ) (correct ✓ ) RHS = [L1] ( sin([T].[T-1] + cos([T].[T-1] ) = [L1] ( sin(M0L1T0) + cos(M0L1T0) ) = [L1] = RHS = equation is dimensionally correct. |
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