The SHM of a particle is given by x(t)=5 cos(2πt+π/4) in MKS units.Calculate the displacement and the magnitude of acceleration of the particle at t=1.5 s.
Answers
Given,
The simple harmonic motion of a particle is given by, x(t) = 5cos(2πt + π/4) in MKS units.
To find,
displacement and magnitude of acceleration of the particle at t = 1.5 sec.
displacement of particle, x(1.5) = 5cos(2π × 1.5 + π/4) = 5cos(3π + π/4) = -5cos(π/4) = -5/√2 m
acceleration of particle, a = d²x(t)/dt² = -20π²cos(2πt + π/4)
= -20π²cos(3π + π/4) = 20π²cos(π/4) = 10√2π²
Therefore displacement of particle is -5√2 m and acceleration of particle is 10√2π² m/s².
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Answer:
- Displacement (x) of the particle is -2.5 √{2} m
- Acceleration (a) of the particle is 10 √{2} π² m/s²
Given:
- Given Equation:- x (t) = 5 cos (2 π t + π / 4)
- Time period (t) = 1.5 seconds
Explanation:
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From given equation,
⇒ x (t) = 5 cos (2 π t + π /4)
Substituting the value of time (t) = 1.5 secs
⇒ x (1.5) = 5 cos (2 π × {1.5} + π /4)
⇒ x = 5 cos (3 π + π / 4)
As we know that the angle (3 π + π / 4) will lie in the third quadrant, where only tan functions are positive.
Therefore,
⇒ cos (3 π + θ) = - sin θ
Substituting in the above equation,
⇒ x = 5 × - sin(π / 4)
⇒ x = - 5 sin (π / 4)
⇒ x = - 5 × 1 / √{2} ∵ [sin(π / 4) =1 / √{2}]
⇒ x = - 5 / √{2}
Rationalizing gives,
⇒ x = (- 5 × √{2}) / (√{2} × √{2})
⇒ x = (- 5 √{2}) / 2
⇒ x = - 2.5 √{2}
⇒ x = - 2.5 √{2} m
∴ Displacement (x) of the particle is -2.5 √{2} m.
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From the given equation we can make out,
⇒ x (t) = 5 cos (2 π t + π /4)
- Amplitude (A) = 5 m
- Angular Frequency (ω) = 2 π
- Phase (∅) = π / 4
Now, as we know standard acceleration formula,
⇒ a = - A ω² sin(ω t ± ∅)
Substituting the values,
⇒ a = - 5 (2 π)² sin (2 π t + π / 4)
⇒ a = - 5 × 4 π² × sin (2 π t + π /4)
We know that the angle (3 π + π / 4) will lie in the third quadrant, where only tan functions are positive.
Therefore,
⇒ sin (3 π + θ) = - cos θ
Substituting in the above equation,
⇒ a = - 5 × 4 π² × - cos (π /4)
⇒ a = -20 π² × - 1 / √{2} ∵ [sin(π / 4) =1 / √{2}]
⇒ a = 20 π² × 1 / √{2}
⇒ a = 20 π² / √{2}
⇒ a = 10 √{2} π²
⇒ a = 10 √{2} π² m/s²
∴ Acceleration (a) of the particle is 10 √{2} π² m/s².
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