Troubleshooting Trigonometry: Finding Coefficient of Friction on a Slope

[Glype11]

This problem deals with finding the coefficient of friction between an object and a slope. It gives an object with mass m, sliding down a hill with a slope of θ at a constant velocity. I got mgsinθ=fksinθ.
However the book shows mgsinθ=fk. The kinetic friction force and the force down the slope are parallel but directed in opposite directions, so where is my error? I included a drawing for reference.

 

[Nathanael]

I'm assuming "fk" is "$F_k$" which is the force of kinetic friction?

The direction of kinetic friction is naturally in the opposite direction as the motion, so what was your reasoning for multiplying it by sin(θ)?

I got mgsinθ=fksinθ.

That can just be simplified to $mg=F_k$

Does $mg=F_k$ make sense to you? That would imply that the angle θ is irrelevant. Does that seem right?

 

[Nathanael]

so where is my error?

No one can really tell you your error unless you take us through the train of thought that led you to your answer (which you didn't explain).How does the fact that it's moving at a constant velocity effect this problem?

 

[Glype11]

No one can really tell you your error unless you take us through the train of thought that led you to your answer (which you didn't explain).


How does the fact that it's moving at a constant velocity effect this problem?

Because it is constant velocity the down slope component of gravity must balance the frictional force. I figured out my error.

 

[HalfLostAndSearching]

First of all, it is important to note that the equation you have derived, mgsinθ=fksinθ, is correct. This is the equation for the forces in the y-direction, where mg represents the force due to gravity and fk represents the kinetic friction force. This equation takes into account the fact that the force of gravity is acting on the object in the downward direction, while the kinetic friction force is acting in the opposite direction, parallel to the slope.

The equation shown in the book, mgsinθ=fk, is also correct, but it is only taking into account the forces in the x-direction. In this case, the only force acting on the object is the kinetic friction force, which is acting in the direction opposite to the motion of the object. This is why the equation only includes the kinetic friction force and not the force due to gravity.

So, to answer your question about where your error may be, it could be that you are not considering the forces in the x-direction when deriving your equation. It is important to always consider all forces acting on the object in both the x and y-directions in order to accurately describe its motion.

Additionally, it is worth mentioning that the coefficient of friction, represented by the variable k, is a constant value and does not change with the slope of the surface. It is dependent on the materials in contact and the roughness of the surface, but not the angle of the slope.

In conclusion, your equation, mgsinθ=fksinθ, is correct and takes into account the forces in the y-direction. The equation shown in the book, mgsinθ=fk, is also correct and only takes into account the forces in the x-direction. Both equations can be used to solve for the coefficient of friction in this scenario.