The motion of the particle is described by the equation y = a sin ωt + b cos ωt. This is a form of the general equation for simple harmonic motion (SHM), which is y = A sin(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant.

To compare the given equation with the general equation for SHM, we can use the trigonometric identity sin(A + B) = sin A cos B + cos A sin B. If we let A = ωt and B = φ, we can rewrite the general equation for SHM as y = A sin ωt cos φ + A cos ωt sin φ.

Comparing this with the given equation, we can see that a = A cos φ and b = A sin φ. Squaring and adding these two equations gives us a^2 + b^2 = A^2 cos^2 φ + A^2 sin^2 φ = A^2 (cos^2 φ + sin^2 φ) = A^2. Taking the square root of both sides gives us A = √(a^2 + b^2).

Therefore, the motion is SHM with amplitude √(a^2 + b^2), which corresponds to option (d).

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The motion of the particle is described by the equation y = a sin ωt + b cos ωt. This is a form of the general equation for simple harmonic motion (SHM), which is y = A sin(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant.

To compare the given equation with the general equation for SHM, we can use the trigonometric identity sin(A + B) = sin A cos B + cos A sin B. If we let A = ωt and B = φ, we can rewrite the general equation for SHM as y = A sin ωt cos φ + A cos ωt sin φ.

Comparing this with the given equation, we can see that a = A cos φ and b = A sin φ. Squaring and adding these two equations gives us a^2 + b^2 = A^2 cos^2 φ + A^2 sin^2 φ = A^2 (cos^2 φ + sin^2 φ) = A^2. Taking the square root of both sides gives us A = √(a^2 + b^2).

Therefore, the motion is SHM with amplitude √(a^2 + b^2), which corresponds to option (d).

Knowee AI · Accepted Answer

The motion of the particle is described by the equation y = a sin ωt + b cos ωt. This is a form of the general equation for simple harmonic motion (SHM), which is y = A sin(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant.

To compare the given equation with the general equation for SHM, we can use the trigonometric identity sin(A + B) = sin A cos B + cos A sin B. If we let A = ωt and B = φ, we can rewrite the general equation for SHM as y = A sin ωt cos φ + A cos ωt sin φ.

Comparing this with the given equation, we can see that a = A cos φ and b = A sin φ. Squaring and adding these two equations gives us a^2 + b^2 = A^2 cos^2 φ + A^2 sin^2 φ = A^2 (cos^2 φ + sin^2 φ) = A^2. Taking the square root of both sides gives us A = √(a^2 + b^2).

Therefore, the motion is SHM with amplitude √(a^2 + b^2), which corresponds to option (d).

The displacement of a particle varies with time according to the relation y = a sin ωt + b cos ωt.(a) The motion is oscillatory but not SHM.(b) The motion is SHM with amplitude a + b.(c) The motion is SHM with amplitude a2 + b2.(d) The motion is SHM with amplitude √(a2 + b2 )

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