To determine the net force on an object of mass [tex]\( m \)[/tex] sliding down an incline with an angle [tex]\( \theta \)[/tex], we need to analyze the forces acting on the object and how they contribute to the net force in the direction parallel to the surface of the incline.
Here is a step-by-step solution:
1. Identify the Forces:
- The gravitational force acting on the object is [tex]\( mg \)[/tex], where [tex]\( g \)[/tex] is the acceleration due to gravity.
- This gravitational force can be divided into two components:
- One component perpendicular to the incline: [tex]\( mg \cos(\theta) \)[/tex]
- Another component parallel to the incline: [tex]\( mg \sin(\theta) \)[/tex]
2. Force Parallel to the Incline:
- The component of the gravitational force that acts parallel to the surface of the incline is [tex]\( mg \sin(\theta) \)[/tex].
3. Normal Force:
- The normal force [tex]\( F_N \)[/tex] exerted by the surface of the incline acts perpendicular to the surface. It does not affect the net force parallel to the surface of the incline directly.
4. Net Force Calculation:
- Since we are interested in the net force parallel to the incline, we focus on the component [tex]\( mg \sin(\theta) \)[/tex]. There are no other horizontal forces (like friction or applied forces) given in the problem that would alter this component.
Therefore, the net force on the object along the surface of the incline is given by:
[tex]\[ F_{\text{net}} = mg \sin(\theta) \][/tex]
Given the options:
A. [tex]\( mg \sin(\theta) \)[/tex]
B. [tex]\( mg \cos(\theta) - F_N \)[/tex]
C. [tex]\( mg \cos(\theta) \)[/tex]
D. [tex]\( mg \sin(\theta) - F_N \)[/tex]
The correct answer is:
[tex]\[ \boxed{A. \, mg \sin(\theta)} \][/tex]
