A particle moves along y-axis according to the equation y=3+4cos(2π/T)t. The motion of the particle is (1) not SHM (2) Oscillatory but not SHMC (3) SHMC (4) Translatory

To determine the nature of the motion of the particle described by the equation \( y = 3 + 4 \cos\left(\frac{2\pi}{T} t\right) \), we need to analyze the characteristics of the equation itself. This equation represents the vertical position \( y \) of the particle as a function of time \( t \), where \( T \) is the period of the motion.

Understanding the Equation

The equation can be broken down into two parts: a constant term and a cosine function. The constant term \( 3 \) indicates that the particle oscillates around the line \( y = 3 \). The term \( 4 \cos\left(\frac{2\pi}{T} t\right) \) represents the oscillatory behavior, where \( 4 \) is the amplitude of the motion, and \( \frac{2\pi}{T} \) is the angular frequency.

Identifying Simple Harmonic Motion (SHM)

Simple Harmonic Motion (SHM) is characterized by the following features:

• The motion is periodic, meaning it repeats itself at regular intervals.
• The restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction.
• The motion can be described by a sine or cosine function.

In our case, the equation \( y = 3 + 4 \cos\left(\frac{2\pi}{T} t\right) \) clearly shows that the particle oscillates in a periodic manner due to the cosine function. The amplitude of \( 4 \) indicates how far the particle moves from its equilibrium position (which is at \( y = 3 \)).

Conclusion on Motion Type

Since the motion is periodic and can be described by a cosine function, we can conclude that the particle exhibits Simple Harmonic Motion (SHM). The presence of the cosine function confirms that the motion is oscillatory and meets the criteria for SHM.

Therefore, the correct answer to the question regarding the nature of the particle's motion is (3) SHMC (Simple Harmonic Motion with Cosine). This indicates that the particle oscillates around a central position with a specific amplitude and period, characteristic of SHM.