How would you derive the formula

{-Ff+Fgx=ma} to show that µs=tanè

Net force=mass*acceleratin

moving force-resistingforce=ma

Now, the resisting force depends often on mu, and sometimes on a slope, the angle.

Without your problem, I can go no further.

To derive the formula µs = tan(θ), where µs is the coefficient of static friction and θ is the angle of incline, we need to start with the equation -Ff + Fgx = ma. Here's the step-by-step process:

Step 1: Break down the forces involved:
- Ff is the force of friction on the object.
- Fgx is the component of gravitational force acting parallel to the incline.
- ma is the total force acting in the system, where m is the mass of the object and a is its acceleration.

Step 2: Split Fgx into its components:
- Fgx can be broken down into two components: Fgx = mg sin(θ) (the parallel component) and Fgy = mg cos(θ) (the perpendicular component).

Step 3: Understand the forces acting on an object on an incline:
- The force of friction, Ff, opposes the motion of the object and can be represented as Ff = µs · N, where µs is the coefficient of static friction and N is the normal force exerted on the object by the incline.

Step 4: Define the normal force, N:
- The normal force, N, is equal to the perpendicular component of the gravitational force, Fgy, acting on the object.

Step 5: Substitute the equations from Steps 3 and 4 into the original equation (Step 1):
- -µs · N + mg · sin(θ) = ma

Step 6: Replace N with mg · cos(θ) (from Step 4):
- -µs · mg · cos(θ) + mg · sin(θ) = ma

Step 7: Divide both sides by m:
- -µs · g · cos(θ) + g · sin(θ) = a

Step 8: Simplify the equation by factoring out g:
- g (-µs · cos(θ) + sin(θ)) = a

Step 9: Solve for µs:
- µs · cos(θ) = sin(θ)
- µs = sin(θ) / cos(θ)
- µs = tan(θ)

Therefore, the derived formula is µs = tan(θ), where µs represents the coefficient of static friction and θ denotes the angle of incline.

To derive the formula μs = tanθ from the equation -Ff + Fgx = ma, we need to first understand the forces involved and their relationship.

In this equation, -Ff represents the force of friction opposing motion, Fgx represents the force of gravity acting in the x-direction, and ma represents the mass times acceleration. The variable µs represents the coefficient of static friction between two surfaces, and θ represents the angle between the force perpendicular to the surface and the normal force.

The force of friction is given by the equation Ff = μsFn, where Fn is the normal force. The normal force is the force exerted by a surface to support an object resting on it and is equal in magnitude and opposite in direction to the force applied by the object on the surface.

The force of gravity can be resolved into two components: Fgx, acting in the x-direction, and Fgy, acting in the y-direction. Fgx is given by the equation Fgx = mg*sinθ, where m is the mass of the object and g is the acceleration due to gravity.

By combining these equations, we can rewrite the given equation as follows:
-μsFn + mg*sinθ = ma

Since the object is in static equilibrium (not moving), the acceleration is zero, which gives us:
-μsFn + mg*sinθ = 0

Solving for μs by isolating it on one side of the equation:
μsFn = mg*sinθ

Dividing both sides of the equation by Fn yields:
μs = (mg*sinθ) / Fn

From basic trigonometry, we know that the ratio of the adjacent side to the opposite side in a right triangle is equal to the tan of the angle. Therefore, we can rewrite the equation as:
μs = tanθ

So, the derived formula is μs = tanθ, which shows the relationship between the coefficient of static friction and the angle of the surface.