the displacement of an oscillating particle is given by y=a sin (bx+ct+d).Then the dimensional formula for (abcd) will be

Question

Answer

Dimensional Formula of the Oscillating Particle Equation

The dimensional formula for the constants a, b, c, and d in the equation y=a sin (bx ct d) can be found by analyzing the units of each term in the equation.

Analysis of the Equation

y represents displacement, which has units of length (L).

a is the amplitude of the wave, which also has units of length (L).

sin is a dimensionless function.

bx has units of radians, since it is the product of an angle and a length.

ct has units of velocity (LT^-1), since it is the product of a speed and time.

d has units of radians, since it is an angle.

Dimensional Formula for a, b, c, and d

Using the analysis above, we can write the dimensional formula for the constants a, b, c, and d as:

Dimensional formula of a = [L]

Dimensional formula of b = [T]^-1

Dimensional formula of c = [L][T]^-1

Dimensional formula of d = [1]

Where [L] represents the dimensional unit for length, [T] represents the dimensional unit for time, and [1] represents a dimensionless unit.

Explanation of the Dimensional Formula

The dimensional formula for a, b, c, and d in the equation y=a sin (bx ct d) indicates the units of measurement for each of the variables. The amplitude (a) and displacement (y) have units of length (L), while the constants b and d are dimensionless. The constant c has units of velocity (LT^-1), which is consistent with the fact that it represents the product of a length and a speed.

Overall, the dimensional formula for the oscillating particle equation provides a way to determine the units of measurement for the various components of the equation, which can be useful in understanding the physical properties of the system being described.

Answer

Dimension of a similar to y = L.                            (bx + ct +d) will be dimensionless                       so dimension of b = L^-1                                                                     c = T^-1.                                                                   d = 0                                            so total dimension M^0 L^0 T^-1

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