Centripetal Force (problems at bottom)
Lesson 1: Background
Centripetal Force is the force required to change the direction of a moving object.
To understand this we first recall
Newtons 1st Law: an object at rest tends to stay at rest and an object in motion tends to continue in straight line motion (unless acted upon by a force)
So how do we get something to "turn"?
We apply a force like this:
This is called the Centripetal Force. Lets see (mathematically speaking) where this comes from we first note that whenever we have a net force, we have an acceleration (from Newtons 2nd Law: F = ma).
So, put a ball on the end of a string and spin it around your head with a constant speed, say 2.0 meters/sec. Now the speed doesnt change, but its direction does
And because of that change in direction we end up with an acceleration! Why? Recall our definition of acceleration:
It says that a change ANY change in velocity results in acceleration. Remember velocity is a vector: it has speed and direction. So if we change speed, we accelerate (or decelerate). This is something were all familiar with. However, if we change direction, we must, by definition, accelerate as well (because a part of the velocity has changed)!
Exactly what does it mean to accelerate when changing direction?
Lets take a second look at that ball
At the bottom of the picture is the expanded equation for acceleration the acceleration is the difference between the velocity at position 2 and the velocity at position 2.
Numerically, there is no difference! Its 2.0 meters/sec always. However, there is a directional difference lets use some vector graphical addition. We create a v1 vector (reverse its direction) and add it to v2:
The result is the vector (delta)v. If we knew the lengths of these vectors we could measure the length of (delta)v and get a numerical result. For now, note that (delta)v is directly proportional to the acceleration!!!!. We only need to divide (delta)v by the time to find it.
More importantly, look at the direction of the accelerated motion Ill paste the (delta)v vector directly onto the picture of the spinning ball, halfway between velocity1 and velocity2:
Its pointed directly at the center! That tells us the acceleration is acting inward (radially inward, to be exact). In fact, its pointing in the same direction as the Force we showed at the beginning of this discussion.
We divide (delta)v by the time, t, to get the acceleration and we draw
Where ac is the "centripetal" acceleration (designated by the small subscript c).
Now, the Centripetal Force, Fc, is found using Newtons 2nd Law by simply multiplying ac by mass:
Fc = mac
Lesson 2: Derivation of centripetal acceleration equation:
We use our circling ball again, as well as our vector addition. However, this time we introduce features from the circle itself, like radius and the "chord":
a)the centripetal acceleration
b)the centripetal force
ac = v2/R
ac = (2.0 m/s)2/(0.7 m)
ac = 5.71 m/s2
b)For centripetal force,
Fc = mac
Fc = (4.0 kg)(5.71 m/s2)
Fc = 23 Newtons
2. The same setup is now spun at 4.0 meters/sec,.
a)find the centripetal acceleration
b)find the centripetal force
c)divide your answer to (b) by 4.45 to find out how many pounds of force
a)As in the first problem,
ac = (4.0 m/s)2/(0.7 m)
ac = 23 m/s2
Fc = (4.0 kg)(23 m/s2)
Fc = 92 N
c)92/4.45 = 21 lbs of force
note that in problem 1, it was only 5.2 lbs of force (23/4.45). By doubling the speed we increase the centripetal force by 4 times!
Why? How did that increase sneak in?