The displacement of a particle executing SHM is given by X = 3 sin [2πt + π/4] , where 'X' is in meter and 't' is in second. The amplitude and maximum speed of the particle is
3 m, 6π ms-1
3 m, 2π ms-1
3 m, 8π ms-1
3 m, 4π ms-1
To find the amplitude and maximum speed of the particle, we are given the equation for displacement in simple harmonic motion (SHM):
\( X = 3 \sin(2\pi t + \frac{\pi}{4}) \)
Step 1: Compare with the general form of SHM:
The general equation for SHM is:
\( X = A \sin(\omega t + \phi) \)
By comparing the given equation with the general form, we can identify the following parameters:
Step 2: Calculate the Maximum Speed (Vmax):
The maximum speed in SHM occurs when the displacement is at its maximum, which is equal to the amplitude \( A \). The maximum speed can be calculated using the formula:
\( V_{\text{max}} = \omega A \)
Substituting the values we found:
\( V_{\text{max}} = (2\pi \, \text{rad/s}) \times (3 \, \text{m}) = 6\pi \, \text{m/s} \)
Therefore, the amplitude and maximum speed of the particle are: 3 m and \( 6\pi \, \text{m/s} \), respectively. The correct option is (A) 3 m, \( 6\pi \, \text{m/s} \).
The displacement of a particle executing Simple Harmonic Motion (SHM) is given by the equation: \[ X(t) = A \sin(\omega t + \phi) \] where:
The given equation is: \[ X = 3 \sin \left(2\pi t + \frac{\pi}{4}\right) \] Here, \( X \) is in meters and \( t \) is in seconds.
By comparing the given equation with the standard SHM equation, we can identify the parameters:
The velocity \( V(t) \) of the particle is the time derivative of the displacement \( X(t) \): \[ V(t) = \frac{dX}{dt} = \frac{d}{dt} [A \sin(\omega t + \phi)] \] \[ V(t) = A\omega \cos(\omega t + \phi) \]
The speed of the particle is the magnitude of the velocity. The maximum speed \( V_{max} \) occurs when the cosine term is equal to \( \pm 1 \). \[ \mathbf{V_{max} = A\omega} \]
Substitute the values of \( A \) and \( \omega \) we found: \[ V_{max} = (3 \text{ m}) \times (2\pi \text{ rad/s}) \] \[ \mathbf{V_{max} = 6\pi \text{ m/s}} \]
Therefore, the amplitude of the particle is 3 m and the maximum speed is 6π m/s.
Comparing this result with the given options:
The correct option is the first one.
When light shines on a metal, electrons can be ejected from the surface of the metal in a phenomenon known as the photoelectric effect. This process is also often referred to as photoemission, and the electrons that are ejected from the metal are called photoelectrons.
According to Einstein’s explanation of the photoelectric effect :
The energy of photon = energy needed to remove an electron + kinetic energy of the emitted electron
i.e. hν = W + E
Where,