Respuesta :
Answer:
[tex]tan \theta = \mu_s[/tex]
Explanation:
An object is at rest along a slope if the net force acting on it is zero. The equation of the forces along the direction parallel to the slope is:
[tex]mg sin \theta - \mu_s R =0[/tex] (1)
where
[tex]mg sin \theta[/tex] is the component of the weight parallel to the slope, with m being the mass of the object, g the acceleration of gravity, [tex]\theta[/tex] the angle of the slope
[tex]\mu_s R[/tex] is the frictional force, with [tex]\mu_s[/tex] being the coefficient of friction and R the normal reaction of the incline
The equation of the forces along the direction perpendicular to the slope is
[tex]R-mg cos \theta = 0[/tex]
where
R is the normal reaction
[tex]mg cos \theta[/tex] is the component of the weight perpendicular to the slope
Solving for R,
[tex]R=mg cos \theta[/tex]
And substituting into (1)
[tex]mg sin \theta - \mu_s mg cos \theta = 0[/tex]
Re-arranging the equation,
[tex]sin \theta = \mu_s cos \theta\\\rightarrow tan \theta = \mu_s[/tex]
This the condition at which the equilibrium holds: when the tangent of the angle becomes larger than the value of [tex]\mu_s[/tex], the force of friction is no longer able to balance the component of the weight parallel to the slope, and so the object starts sliding down.
The friction force is a type of opposition force act on the body surface are in contact. The expression for the friction force will be [tex]\rm{tan\theta = u_s}[/tex].
What is the friction force?
The friction force is a type of opposition force act on the body surface are in contact.
when an object is moving on a rough surface the object experience some type of force that wants to oppose the motion of the body is known as friction force.
It is given by the product of the coefficient of friction and the normal force acting on the body.
f = µN
If the net force acting on an item is zero, it is at rest along a slope. The forces in the direction parallel to the slope have the following equation:
[tex]mgsin\theta - u_sR = 0[/tex]
mg sinθ is the weight component parallel to the slope,
where m is the mass of the item
g is the acceleration of gravity,
θ is the slope angle.
The forces along the slope's perpendicular direction have the following equation:
[tex]\rm{R - mgcos\theta = 0}[/tex]
R stands for normal reaction
mg is the weight component perpendicular to the slope
[tex]\rm{R = mgcos\theta[/tex]
substitute the values
[tex]\rm{mgsin\theta - u_smgsin\theta = 0}[/tex]
[tex]\rm{mgsin\theta = u_smgsin\theta}[/tex]
[tex]\rm{tan\theta = u_s}[/tex]
Hence the obtained expression for the friction force will be [tex]\rm{tan\theta = u_s}[/tex]
To know more about friction force refer to the link;
https://brainly.com/question/1714663
