# Derivation of equation for sliding object

• hitek0007
In summary, the equation for acceleration of a sliding object on a ramp is a=g(sin\theta-\mu cos\theta), where \theta is the ramp angle and \mu is the coefficient of friction. As the ramp angle increases, the acceleration also increases. The mass of the object does not affect the acceleration in this equation.
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.

a=Fnet/m
Fgramp=mgsin(x)
Ffk=ukmgcos(x)

## The Attempt at a Solution

a=Fnet/m
a=(Fgramp-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!

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.

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.

## 1. What is the equation for a sliding object?

The equation for a sliding object is F = μN, where F is the force of friction, μ is the coefficient of friction, and N is the normal force.

## 2. How is the equation derived?

The equation is derived from the laws of friction, which state that the force of friction is proportional to the normal force and the coefficient of friction. It is also based on the assumption that the object is sliding at a constant velocity.

## 3. What is the significance of the coefficient of friction?

The coefficient of friction is a dimensionless value that represents the roughness or smoothness of the surfaces in contact. It determines the amount of force required to move an object over a surface.

## 4. Can the equation be used for all types of sliding objects?

The equation is commonly used for objects sliding on a flat surface, but it can also be applied to objects sliding on an incline or other surfaces. However, the coefficient of friction may vary depending on the surface and conditions.

## 5. Are there any limitations to the equation?

The equation assumes ideal conditions, such as a constant velocity and a flat surface. It may not accurately predict the force of friction in real-world scenarios where there are other factors at play, such as air resistance and surface imperfections.

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