# An object with a mass of #6 "kg"# is on a ramp at an incline of #pi/8 #. If the object is being pushed up the ramp with a force of # 2 "N"#, what is the minimum coefficient of static friction needed for the object to remain stationary?

##### 1 Answer

#### Explanation:

**Step 1: Draw a free body diagram.**

Draw an object on a ramp inclined at an angle of *static* friction

The friction acts parallel to the slope but it is ok at this point to know whether it points upwards or downwards. Whether it points up the ramp or down the ramp depends on how

**Step 2: Use the equilibrium condition.**

From Newton's Second Law (

**Step 3: Choosing a smart coordinate system.**

One way to perform vector addition is to resolve all the vectors into their individual components first, then add up the resolved vectors. Rather than using the convention that

To be more precise, we arbitrarily assign the bottom of the ramp to be the origin, it does not matter at all.

**Step 4: Adding the forces.**

To achieve equilibrium, both the

Therefore,

The static friction required to hold the object points up the ramp.

Now for the

Therefore,

**Step 5: Finding the minimum coefficient of static friction.**

For the object not to slip,

where

Rearranging yields

The minimum coefficient of static friction is about 0.377.