Good morning. We already discussed the Non-uniform Circular Motion of a ball on the end of a string.
Today we are going to discuss the Non-uniform Circular Motion of this accelerating toy car.
♪ Flipping Physics ♪ Okay.
Let’s slow the video down so we can more easily discuss what is happening. Bo, tell me what you
notice about the tangential velocity of the car which is speeding up and turning, and how that
affects the linear accelerations of the car. Well, initially the car is at rest. So,
the initial tangential velocity of the car is zero. And, as the car speeds up,
the magnitude of the tangential velocity of the car increases. So, because the magnitude of the
tangential velocity is increasing, the car must be experiencing a tangential acceleration which is
in the same direction as the tangential velocity. (Very nice Bo. Now, it’s probably not quite true,
however, for simplicity’s sake, we are going to assume the tangential acceleration of this
car is constant. Bobby, what about the centripetal acceleration of the car?)
Okay. The center-seeking, centripetal acceleration of the car. Well, because the direction of the
tangential velocity is changing, there must be a centripetal acceleration which is directed inward.
Actually, previously we derived the equation for the magnitude of the centripetal acceleration
of an object moving along an arc to equal tangential speed squared over radius. Therefore,
because the initial tangential speed of the car is zero, the initial centripetal acceleration
of the car is zero. And as the tangential speed of the car increases, the centripetal
acceleration of the car must also increase. Thank you Bobby. Now, again, for the sake of
simplicity, we are going to assume the radius of the arc the car is moving through is constant. At
this point notice the car has both a tangential acceleration directed tangent to the arc the car
is moving through and a centripetal acceleration which is directed inward along the radius of the
arc the car is moving through, and those two accelerations are perpendicular to one another.
And therefore, these two accelerations are the components of the acceleration of the car. Any
object moving in non-uniform circular motion will have these two components of acceleration:
tangential and centripetal acceleration. And the acceleration of the object will be the
vector sum of the tangential and centripetal accelerations. Billy, if we knew the value of
the tangential and centripetal accelerations of this car at a particular location,
please tell me how we would determine the magnitude and direction of the car’s acceleration.
Absolutely. Uh to find the magnitude of the acceleration of the car, we would use the
Pythagorean Theorem or tangential acceleration squared plus centripetal acceleration squared
equals acceleration squared. Which would mean that the magnitude of the acceleration of the
car would equal the square root of the quantity tangential acceleration squared plus centripetal
acceleration squared. To find the direction of the acceleration, define theta as the
angle between the tangential acceleration and acceleration. Tangent of theta equals opposite
over adjacent or centripetal acceleration over tangential acceleration. Therefore,
theta equals the inverse tangent of centripetal acceleration over tangential acceleration.
Thank you Billy. Now let’s look at the force causing the acceleration of the car.
The only force acting on the car in the horizontal plane the car is moving in is the force of static
friction, therefore, according to (Hold up mr.p.)
Yes, Bo? Static friction? The car
is moving, so how can it be static friction? The tires are rolling without slipping , so
it’s static friction. Okay, that makes sense.
But how is a force of friction causing the car to move? I thought it was a force applied that moved
a car. You “apply” your foot to the accelerator pedal, and that accelerates the car, right?
As mr.p just said, the only force acting on the car in the horizontal plane the car is moving in
is the force of static friction. I mean, is there anything external to the car which is pushing on
the car to cause a force applied on the car? No, but I just always thought a force applied
caused a car to accelerate. Actually no. Sometimes we have
simplified it that way, however, it is actually the force of friction which accelerates a car.
But aren’t there 4 tires so 4 forces of static friction acting on the car?
Ah, good point Bo. We are using a simplified model for this. In other words, we are modeling the
motion as one force of static friction acting on the car instead of 4 forces of static friction, 1
for each tire. This simplified model with 1 force of static friction is fine for the Non-uniform
Circular Motion we are discussing today. Hold up. Isn’t there also a force of gravity
acting down on the center of mass of the car and 4 normal forces acting up on the car,
ya know, 1 normal force for each tire? Remember, we are only concerning ourselves with
the forces which act in the horizontal plane the car is moving in because there is no acceleration
perpendicular to the horizontal plane. Oh yeah. Right. Thanks.
Okay, let’s try again. The only force acting on the car in the horizontal plane the car
is moving in is the force of static friction, therefore, according to Newton’s Second Law,
the forces which are causing the tangential and centripetal accelerations are the components of
the Force of Static Friction. In other words, the net force in the in-direction equals the
component of the force of static friction in the in-direction which equals mass times centripetal
acceleration. And the net force in the tangential direction equals the component of the force
of static friction in the tangential direction which equals mass times tangential acceleration.
That means the components of the Force of Static Friction are proportional to the components of
the acceleration. And, the components of the force of static friction add up to the
force of static friction. In other words, the force of static friction starts tangent to the
circle where the tangential speed and centripetal acceleration are both zero, and then increases in
magnitude and rotates its direction such that it slowly acts more and more in an inward direction.
Got it. Okay. Sure. The main takeaways from this lesson
are: Non-uniform Circular Motion has both a tangential and centripetal component to its linear
acceleration, therefore, the linear acceleration of the object acts in a direction between
the tangential and inward directions. And, in this example, linear acceleration is caused by
the force of static friction which acts in the same direction as the acceleration of
the car. Thank you very much for learning with me today, I enjoy learning with you.