Question

If \(x=5\sin(\pi i+\frac{\pi}{3})m\) represents the motion of a particle executing simple harmonic motion, the amplitude and time period of motion, respectively, are :

• A. 5 cm, 2 s
• B. 5 m, 2 s
• C. 5 cm, 1 s
• D. 5 m ,1 s

Answer 1

The Correct Option is B

Concepts: Simple harmonic motion, Amplitude, Time period

Explanation:

The given equation is x = 5sin(πt + π/3) m. This represents a simple harmonic motion (SHM) equation of the form x = A sin(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant.

From the equation, we can see that:

The amplitude A is the coefficient of the sine function, which is 5 meters.

The angular frequency ω is the coefficient of t inside the sine function, which is π.

The time period T of the SHM is given by the formula T = 2π/ω. Substituting ω = π: T = 2π/π = 2 seconds.

Thus, the amplitude is 5 meters and the time period is 2 seconds.

Step by Step Solution:

Step 1: Identify the amplitude A from the SHM equation x = A sin(ωt + φ). Here, A = 5 meters.

Step 2: Identify the angular frequency ω from the SHM equation. Here, ω = π.

Step 3: Use the formula for the time period T = 2π/ω.

Step 4: Substitute ω = π into the formula to find T = 2π/π = 2 seconds.

Final Answer:

The amplitude is 5 meters and the time period is 2 seconds.

Answer 2

Equation for Simple Harmonic Motion (SHM)

The equation for simple harmonic motion (SHM) is given by:

$$ x = A \sin(\omega t + \phi) $$

Where:

x = Displacement at time \( t \)

A = Amplitude

\(\omega\) = Angular frequency

t = Time

\(\phi\) = Phase constant

Step 1: Compare with the Given Equation

In the given equation:

$$ x = 5 \sin (\pi t + \frac{\pi}{3}) $$

Comparing with the standard form:

Amplitude (\( A \)) = 5 m

Angular frequency (\( \omega \)) = \( \pi \)

Step 2: Find the Time Period

The time period \( T \) is related to the angular frequency \( \omega \) by the formula:

$$ \omega = \frac{2\pi}{T} $$

Substituting \( \omega = \pi \):

$$ \pi = \frac{2\pi}{T} $$

Solving for \( T \):

$$ T = \frac{2\pi}{\pi} = 2 \text{ s} $$

Conclusion

The amplitude is 5 m, and the time period is 2 s.