centripetal-forcefrictionhomework-and-exercisesnewtonian-mechanics
I don't understand how the force of friction can face a different direction in each case. To my understanding, without friction there is a net force down the slope. We want the net force to be towards the right (centripetal acceleration). So why isn't the force of friction always up the slope to cancel the y component of the net force?
I had fun trying to make this as intuitive as possible. I hope I've succeeded without doing the physics of the situation much injustice.
When a car is driving straight ahead, the plane in which the wheels are rotating is aligned with the direction of movement. Another way of saying this is that the rotation axis is perpendicular to the momentum vector $\vec{p}=m\vec{v}$ of the car. So the friction merely makes it harder for the car to move, which is part of the reason why you need to put your foot on the gas pedal to maintain a constant speed. At the same time, the friction is what allows you to maintain that constant speed because the rotating tires sort of grab onto the ground, which is the intuitive picture of friction. The tires grab the ground and pull/push it backwards beneath themselves, as you would do when dragging yourself over the floor (if it had handles to grab onto). Those grabbing and pulling/pushing forces are what keeps you going.
Things change when the wheels are turned. The plane in which they are rotating now is at an angle with the direction of motion. Alternatively but equivalently, we could say the rotation axis now makes an angle with the momentum vector of the car. To see how friction then makes the car turn, think again in terms of the wheels grabbing onto the ground. The fact that they now make an angle with the direction of motion, means the force the tires are exerting is also at an angle with the direction of motion - or equivalently, the momentum vector.
Now, a force is a change in momentum$^1$ and so (because the wheels are part of the rigid body that is a car) this force will change the direction of the car's momentum vector until it is aligned with the exerted force. Imagine dragging yourself forward on a straight line of handles on the floor and then suddenly grabbing hold of a handle slightly to one side instead of the one straight ahead. You'll steer yourself away from the original direction in which you were headed.
$^1$ Mathematically: $$\vec{F}=\frac{d\vec{p}}{dt}$$
This is my attempt to illustrate what happens when the car wheel is turned:
Focus on the bit of the car tyre marked with a red spot, and the bit of the road marked with a green spot. If we could look at the contact patch between the tyre and the road we'd see something like the rectangle I've drawn on the left. When the wheel is straight the red spot on the tyre and the green spot on the road move together (I'm assuming the car is moving to the left so the ground is moving to the right).
but now suppose we turn the wheel to the left. The contact patch now looks like the rectangle on the right. Because the patch has been rotated relative to the road the red and green spots now don't move together, but instead the red spot on the tyre is scraped across the road surface. It's this lateral motion of the tyre surface across the road surface that causes the frictional force that turns the car.
Best Answer
You o want the net force to be to the right, but when the component of gravity that is to the right is already larger than that, then you need friction to point the other way. So the free-body diagrams are correct, all that remains is to use the v and r to figure out the centripetal force, and then make friction, gravity, and the normal force add up to it.