An object of mass $$m$$ is being moved with a constant velocity under the action of an applied force of 2 N along a frictionless surface with following surface profile.

The correct applied force vs distance graph will be:

We need to determine the correct graph showing the applied force ($$F$$) vs. distance ($$x$$) required to move an object at a constant velocity along a frictionless surface profile.

1. Understand the Physics of Constant Velocity

When an object moves with a constant velocity, its acceleration is zero ($$a = 0$$). According to Newton's second law, the net force acting on the object along the direction of motion must be zero:

$$\Sigma F = 0 \implies F_{\text{applied}} + F_{\text{gravity, parallel}} = 0$$

$$F_{\text{applied}} = -F_{\text{gravity, parallel}}$$

This means the applied force must exactly balance the component of the gravitational force acting along the slope at every point on the surface profile.

2. Analyze the Surface Profile Phases

Though the exact slope values depend on the specific geometry , let's look at the standard behavior for a symmetric dip or hill of length $$D$$:

• First Half of the Profile ($$0$$ to $$\frac{D}{2}$$):
  • If the profile slopes downward, gravity pulls the object forward ($$+F_{\text{gravity}}$$). To maintain a constant velocity, the applied force must pull backward to resist it, making it negative ($$F = -2\text{ N}$$).
  • If the slopes are straight planes, the gravitational component remains constant, resulting in a flat, constant negative force value.
• Second Half of the Profile ($$\frac{D}{2}$$ to $$D$$):
  • As the object moves up the opposing incline, gravity pulls it backward ($$-F_{\text{gravity}}$$). To keep moving forward at a constant velocity, the applied force must push forward, making it positive ($$F = +2\text{ N}$$).
  • For a uniform linear slope, this positive force remains constant throughout this zone.
• Beyond Distance $$D$$:
  • Once the object returns to the flat horizontal surface, there is no component of gravity acting along the surface ($$F_{\text{gravity, parallel}} = 0$$). On a frictionless surface, no applied force is needed to sustain constant velocity, so $$F = 0$$.

3. Match with the Graphs

• Graph A: Shows a sudden step transition. It is constant and negative ($$-2\text{ N}$$) for the first section, switches to a constant positive value ($$+2\text{ N}$$) for the second section, and drops back to zero past distance $$D$$. This perfectly matches a surface profile composed of straight, flat-facetted inclines (like a V-shaped or wedge-shaped valley).
• Graph B: Shows a continuously changing force forming a triangular shape, which would correspond to a continuously changing curved profile (like a smooth parabolic hill or depression).

Conclusion

The correct representation of the step-like force variation for flat-facetted inclines is shown in Graph A (Option A).