Derivation of equation for sliding object

[hitek0007]

Homework Statement

1. The problem:

Derive an expression in its simplest form to show the relationship between the mass, ramp angle, and acceleration of the sliding object. Explain clearly but briefly the effect of each term in the expression on the actual acceleration, as the ramp angle changes.

No known variables. Context of this question: part of a lab, in which we found values of friction and coefficients of friction through measuring acceleration of objects sliding down a ramp.

Homework Equations

a=F_net/m
Fg_ramp=mgsin(x)
Ff_k=u_kmgcos(x)

The Attempt at a Solution

a=Fnet/m
a=(Fg_ramp-Fk)/(m)
a=(mgsin(x)-ukmgcos(x))/(m)
a=gsin(x)-ukgcos(x)
a=g(sin(x)-ukcos(x))

Is this the right equation? If so, mass has no effect on the acceleration. Acceleration increases as angle increases.

However, my teacher told me that the equation is supposed to look like:

a=_________+_________
I suppose it is possible the 2nd term is negative... but I am not sure.

Are there different equations? We also calculated ideal and measured accelerations to find the value of friction. Is this any use?

Thanks!

 

[americanforest]

Looks like you've got it right.

$$a=g(sin\theta-\mu cos\theta)$$ where $$\theta$$ is the angle between the ramp and the horizontal. Note that $$a$$ is the acceleration when the object has been released and slides down the ramp. If the object has been pushed up the ramp and is in the process of sliding up then a slightly different equation governs its acceleration.

 

[thomate1]

Your derivation of the equation for the sliding object looks correct. The first term, gsin(x), represents the component of the object's weight that is parallel to the ramp and contributes to its acceleration. The second term, -ukgcos(x), represents the force of friction, which acts in the opposite direction and decreases the acceleration of the object. As the ramp angle increases, the component of weight parallel to the ramp also increases, resulting in a higher acceleration. However, as the ramp angle increases, the force of friction also increases, leading to a decrease in acceleration.

There may be different equations for this scenario, depending on the specific variables and assumptions used. It would be helpful to clarify what the ideal and measured accelerations were used for in your calculations. These values may provide additional insight into the relationship between the variables and could potentially be used to validate or refine the equation.