The question asks about the nature of the restoring force in Simple Harmonic Motion (SHM). We need to recall the defining characteristics of SHM, particularly how the restoring force relates to displacement.
C) Directly proportional to displacement and directed towards mean position
This option perfectly matches the definition and mathematical representation of the restoring force in Simple Harmonic Motion, \(F = -kx\).
The question asks to identify the term for the quantity \((\omega t + \varphi)\) in the given displacement equation for Simple Harmonic Motion (SHM). We need to recall the standard form of the SHM equation and the definitions of its components.
C) The quantity \((\omega t + \varphi)\) is called the phase. It represents the instantaneous state of the oscillation at time \(t\).
This question asks about the relationship between angular frequency (\(\omega\)) and time period (\(T\)) in Simple Harmonic Motion (SHM). Understanding the definitions of these terms is key to deriving their relationship.
This formula directly connects the angular frequency to the time period of the oscillation.
A) \(\omega = 2\pi/T\) is the correct relationship between angular frequency and time period in Simple Harmonic Motion, as derived from their definitions.
The question asks about the characteristics of acceleration in Simple Harmonic Motion (SHM). To answer this, we need to recall the defining equation of SHM and understand the relationship between acceleration, displacement, and the direction of acceleration.
C) Proportional to displacement and directed towards mean position. This statement accurately describes the acceleration in SHM, as derived from the defining equation \(a = -\omega^2 x\).
To find the maximum velocity of a particle executing Simple Harmonic Motion (SHM), we need to recall the general equation for displacement in SHM and then differentiate it with respect to time to get the velocity equation. The maximum velocity occurs when the sine or cosine term in the velocity equation is at its maximum value (which is 1).
Displacement in SHM: The displacement of a particle executing SHM can be represented by the equation:
\[ x(t) = A \sin(\omega t + \phi) \]where:
Velocity in SHM: Velocity is the rate of change of displacement with respect to time. So, we differentiate the displacement equation:
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt} [A \sin(\omega t + \phi)] \]\[ v(t) = A \omega \cos(\omega t + \phi) \]Maximum Velocity: The velocity \(v(t)\) will be maximum when the \(\cos(\omega t + \phi)\) term is at its maximum value. The maximum value of \(\cos(\theta)\) is \(1\).
Therefore, the maximum velocity, \(v_{\text{max}}\), is:
\[ v_{\text{max}} = A \omega (1) \]\[ v_{\text{max}} = A \omega \]D) \(\omega A\) is the correct expression for the maximum velocity of a particle executing SHM.
To find the maximum acceleration of a particle executing Simple Harmonic Motion (SHM), we need to recall the general equation for displacement in SHM and then differentiate it twice with respect to time to obtain the acceleration. The maximum value of this acceleration will depend on the amplitude and angular frequency.
Displacement in SHM: The displacement of a particle executing SHM can be represented by the equation:
\[ x(t) = A \sin(\omega t + \phi) \]where \(A\) is the amplitude, \(\omega\) is the angular frequency, \(t\) is time, and \(\phi\) is the initial phase angle.
Velocity in SHM: To find the velocity, we differentiate the displacement equation with respect to time:
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt} [A \sin(\omega t + \phi)] = A \omega \cos(\omega t + \phi) \]Acceleration in SHM: To find the acceleration, we differentiate the velocity equation with respect to time:
\[ a(t) = \frac{dv}{dt} = \frac{d}{dt} [A \omega \cos(\omega t + \phi)] = -A \omega^2 \sin(\omega t + \phi) \]Maximum Acceleration: The acceleration \(a(t)\) is maximum when the sine term \(\sin(\omega t + \phi)\) is at its maximum absolute value, which is \(\pm 1\). Therefore, the magnitude of the maximum acceleration, \(a_{\text{max}}\), is:
\[ a_{\text{max}} = |-A \omega^2 (\pm 1)| = A \omega^2 \]D) \(\omega^2 A\) is the correct expression for the maximum acceleration of a particle executing SHM.