# Kinetic friction

If there is slipping between surfaces, then the kinetic frictional force is given by where is the coefficient of kinetic friction, N is the normal force, and the direction of the force is parallel to the surface in opposition to the slipping. In general is less than so once an object is moving, the force of friction is less than the maximum friction when the object is still. |

*not*true. (It is true for air resistance, but not friction.)

**Figure 6-7**

Brad pushes a stove (100 kg) in a straight path across the level floor at constant speed 0.2 m/s. The coefficient of kinetic friction is 0.3 for the stove and the floor. What is the force that Brad must apply?

Figure 6-8a |
Figure 6-8b |

Figure 6-8c |
Figure 6-8d |

First, we DRAW A DIAGRAM (Figure 6-8a). The words “constant speed” and “straight path” should send bells off in our head. There is no acceleration, so the vertical equation becomes,

*N* – *mg* = (*F*_{net})_{y} = 0

*N* = *mg* = (100 kg)(10 m/s^{2}) = 1000 N

Equation (2) gives the friction

*F*_{k}= *µ*_{k}*N* = 0.3(1000 N) = 300 N

The horizontal equation becomes

*F*_{B} – *F*_{k} = (*F*_{net})_{x} = 0

*F*_{B} = *F*_{k} = 300 N

Think about this. Brad’s pushing force is equal in magnitude to the frictional force.

“But wait a minute!” some readers will cry. “Doesn’t Brad have to *overcome* the force of friction for the stove to be moving? Brad’s force must be *greater* than the frictional force!” But that is exactly not the case. If the stove is moving at *constant speed*, then the forces must balance. If Brad exceeded the force of friction, the stove would be accelerating.

Perhaps it would help if we looked at the whole Brad/stove story. When Brad approaches the stove, the force diagram on the stove looks like Figure 6-8b. As Brad begins to push on the stove, the friction vector gets larger, as in Figure 6-8c. The moment Brad exceeds *F*_{s,max}, the stove budges, and the force of friction shrinks from *F*_{s,max} to *F*_{k}. Now there is a net force on the stove, and it accelerates from rest (Figure 6-8d). Once the stove is moving, it gets away (a little) from his hands, and Brad’s force decreases to become *F*_{k}. At this point the stove has attained some constant speed, which it keeps. See Figure 6-8a, where the two horizontal vectors are equal in magnitude.

A student is pushing a chalk eraser (0.1 kg) across a level desk by applying a force 0.3 Newtons at an angle directed downward, but 30˚ from the horizontal. The eraser is moving at constant speed 0.1 m/s across the desk (in a straight line). What is the coefficient of friction between the eraser and the desk?

First, we DRAW A DIAGRAM (Figure 6-9). Constant velocity tells us that (*F*_{net})_{x} and (*F*_{net})_{y} are zero. The vertical equation becomes

*N* – *T*_{y} – *mg* = (*F*_{net})_{y}

*N* = *T*_{y} + *mg* = *T*sin 30 + *mg*

*N* = (0.3 N)sin 30 + (0.1 kg)(10 m/s^{2})

*N* = 1.15 N.

The horizontal equation becomes

*T*_{x} – *F*_{k} = (*F*_{net})_{x} = 0

*F*_{k} = *T*_{x} = *T*cos 30 = 0.26 N

Thus we can calculate *µ*_{k}

*µ*_{k} = *F*_{k}/*N* = 0.26N/1.15N = 0.23

**Figure 6-9**

A car (1000 kg) is traveling downhill at 20 m/s in the rain. The grade of the road is 20%, which means that for every 100 meters of road, the vertical drop is 20 meters. The driver sees Bambi in the road and slams on the brakes. The coefficient of kinetic friction between the tires and the road is 0.5. How much time does it take the car to skid to a halt?

(Hint: If *θ* is the angle between the horizontal and the road, then cos*θ* = 0.98 and sin*θ* = 0.2. Also *g* = 10 m/s^{2}.)

Here we will merely sketch a solution. You should try to work out the details. Figure 6-10 shows the force diagram. Working out the vertical equation with (*F*_{net})_{y} = 0 gives

*N* = 9800 N.

Kinetic friction is then

*F*_{k} = 4900 N.

Working out the horizontal equation (there is a net force) gives

(*F*_{net})_{x} = 2900 N

*a*_{x} = 2.9 m/s^{2}

Using the acceleration, the initial velocity, and the final velocity *v*_{2x} = 0, we obtain

∆*t* = 7 s.

(Bambi was unscathed, but only because he jumped off the road in time.)

**Figure 6-10**