# Homework Help: Object on a plane (equilibrium problem)

1. Aug 24, 2010

### ZacBones

1. The problem statement, all variables and given/known data

A 1350kg car is at rest on a plane surface.
The unit vector normal to the surface is: 0.231i + 0.923j + 0.308k.
The y axis points upwards.

Find the magnitude of the normal and friction forces the car's wheels exert on the plane.

2. Relevant equations

$$\sum$$ F = 0.

3. The attempt at a solution

--The weight has the vector W = 1350*9.81i
--The normal force has the vector FN = 0.231|FN|i + 0.923|FN|j + 0.308|FN|k

I also need to express the frictional force as a vector. I know I can find an infinite number of vectors normal to the given vector, but I don't see how I am to choose which is the right one. Mentally, I imagine the car tending to slide "down" the slope relative to its incline, but I'm not sure how to turn this intuition into a vector involving the friction force. Moreover, wouldn't the slide direction depend on the relative strength of the gravitational vs. the normal force?

Even if I found that, I would still need to solve a system of three equations (resulting from setting the sum of components of the force vectors equal to zero). This makes me question my strategy since there are only two unknowns.

I feel like I am missing something obvious here, or that I'm over-complicating the problem. Can anyone clue me into where I am going wrong?

Thanks,
-Zac

2. Aug 25, 2010

### ehild

First: the y axis points upward, and the unit vector in the y direction is usually called j. So the weight is W = 1350*9.81 j.

The weight has a component normal to the surface, parallel with the normal of the plane. You can find this component from the scalar product of the weight with the normal vector. This component cancels with FN, the force between the car and the surface.

The other component of the weight is parallel with the slope. You need a frictional force of equal magnitude and opposite sign to cancel it.

ehild

3. Aug 25, 2010

### ZacBones

Thanks for the reply. Typically I solve these sorts of problems by solving the system of component equations summed to zero...it didn't occur to me to split the forces the way you suggested.

It also took me forever to remember how to compute the gradient of a plane and then find a line parallel to the plane and following the gradient, but once I did, I was *finally* able to solve for the frictional force.

(Next time I ask something I'll use angle-bracket notation instead of i/j/k in order to avoid typos.)

Thanks,
-Zac