Active Figure 4.21 Acceleration Up and Down an Incline with Friction
The external forces exerted on a block lying on a rough incline are the gravitational force
mg, the normal force
n, and the force of friction
f. For convenience, the gravitational force is here is resolved into a component along the incline and a component perpendicular to the incline.
Instructions:
Turn friction on or off using the drag button and click start to project the block up the incline. You can use the back, pause and fwd buttons to stop the block's motion at any point and step through one frame at a time.
Exploration 1
For this Exploration, set the drag button to off. Reset the the block and start a sequence. Observe how the block rises to a highest point and then moves down the slope out of view. Reset the block and start another sequence, but this time use the pause, back and fwd button to freeze the block at its highest point. Record the time, distance and acceleration at this highest point. The acceleration graph in the upper right-hand corner of the screen shows that the acceleration is constant throughout the entire motion sequence. Why?
(a) From your results recorded above, find the following quantities.
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Exploration 2
Click the drag button to introduce a kinetic frictional force on the block. Reset the block and run a full sequence. Notice in this case that the acceleration varies from a maximum value of 8.61 m/s
2 to a minimum value of 1.19 m/s
2.
To understand why the acceleration now changes, play the animation a few times, paying close attention to the free-body diagram. Use the pause button to play a few frames right around the highest point. Recall that friction always acts opposite to the direction of slide. So, while the block slides up the incline, the friction points down the incline -- and vice-versa.
To find the magnitude of the friction force, apply Newton's second law in a direction perpendicular to the incline.
Fy =
-mg cos

= 0
fk = µ
kn = µ
kmg cos
While the block is sliding up the incline (negative direction), Newton's second law applied in a direction parallel to the incline gives the following.
Fx = sin(

) +
fk =
maup
However, when the block slides down the incline, Newton's second law becomes:
Fx =
mg sin

-
fk =
madown
(b) Use either of the last two equations to solve for the coefficient of kinetic friction. Double-check with the other equation.
µ
k =
Exercise 4.AF.21
Suppose that the coefficient of kinetic friction in the above animation is decreased to 0.267. Calculate the magnitude of acceleration up and down the incline.