Understanding SHM and Potential Field
A particle of mass \( m \) undergoing simple harmonic motion (SHM) in a potential field described by \( U(x) = cx^2 \) experiences a restoring force proportional to the displacement from the equilibrium position. Here, \( c \) is a positive constant.

Characteristics of the Potential Field
- The potential energy \( U(x) = cx^2 \) defines a parabolic shape.
- The restoring force \( F \) is given by \( F = -\frac{dU}{dx} = -2cx \).
- This indicates that the force is linearly proportional to the displacement \( x \), characteristic of SHM.

Expression for Time Period
The time period \( T \) of a particle in SHM is given by:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
where \( k \) is the effective spring constant.
In our case, we identify \( k \) from the relation:
\[ k = 2c \]
Thus, substituting this into the formula for \( T \):
\[ T = 2\pi \sqrt{\frac{m}{2c}} \]

Dependence on Constant \( c \)
- The time period \( T \) is inversely proportional to the square root of the constant \( c \).
- As \( c \) increases, \( T \) decreases, indicating that the motion becomes faster.

Graphical Representation
- The graph of \( T \) versus \( c \) is a hyperbolic curve.
- It demonstrates that \( T \) decreases as \( c \) increases, which showcases the relationship between time period and the stiffness of the potential.

Conclusion
In conclusion, the time period of a particle in SHM under the influence of a potential field \( U(x) = cx^2 \) is expressed as:
\[ T = 2\pi \sqrt{\frac{m}{2c}} \]
This relationship highlights the effect of the potential constant \( c \) on the particle's motion.