Type: Double
Date: ______________
Objective: Vertical Circular Motion II
Vertical Circular Motion II
Homework: Read 5.7 Do CONCEPT QUEST # (14, 15),
Do PROBLEMS # (36, 54) Ch. 5
+ 1984 #1 (handout)
AP Physics “B”
Mr. Mirro
Date: ________
Vertical Circular Motion I
Motorcycle stunt drivers often perform a feat in which they drive their cycles around a vertical circular
track as shown in figure (a). Usually, the speed varies in this stunt.
When the speed of travel on a circular path changes from
moment to moment, the motion is said to be “non-uniform.”
Nonetheless, we can use the concepts that apply to uniform
circular motion to gain considerable insight into the motion
that occurs on a vertical circle.
There are four (4) points on a vertical where the centripetal force can
be identified easily as shown in figure (b).
If we consider Newton’s Second Law of Acceleration:
∑ Fc = m ac
Keep in mind that:
‰
Centripetal force is NOT a new and separate force of
nature.
‰
At each point the centripetal force is the SUM of ALL
the force components oriented along the “radial” direction
and points toward the center of the circle.
As the cycle goes around, the magnitude of the normal force changes. It changes because the speed
changes and because the weight does not have the same effect at every point.
‰
At the bottom, position (1), the weight and the normal force “oppose” each other.
∑ Fc = m ac
That is,
‰
+ FN1 – mg = m ac ⇒ + mv2
r
At the top, position (2), the opposite is true. The weight and the normal force “reinforce” each
other.
∑ Fc = m ac
That is,
- FN2 – mg = - m ac ⇒ - mv2
r
At points (2) and (4) on either side, only the normal force provides the centripetal force. The weight is
tangent to the circle and has NO component pointing toward the center !
Therefore,
‰
On the right, at position (2) Newton’s Law goes as:
∑ Fc = m ac
That is,
- FN2 – mg = - m ac ⇒ - mv2
r
- FN2 ⇒ - mv2
r
So,
‰
FN2 ⇒ mv2
r
On the left, at position (4) Newton’s Law goes as:
∑ Fc = m ac
That is,
- FN4 – mg = - m ac ⇒ - mv2
r
- FN4 ⇒ - mv2
r
So,
FN4 ⇒ mv2
r
Riders who perform the loop-the-loop trick know that they must have AT LEAST a minimum speed at
the top of the circle to remain on the track.
‰
This speed can be determined by considering the centripetal force at point (3).
‰
At this speed, the track does not exert a normal force to keep the cycle on the circle, because the
weight (mg) provides all the necessary centripetal force.
AP Physics “B”
Mr. Mirro
Date: ________
Vertical Circular Motion I
Ex 1: A fighter pilot dives her plane toward the ground at nearly Mach-1 (Vsos ≈ 331 m/s @ STP). She
pulls out of the dive on a vertical circle as shown. Neglecting the mass of her plane, what is the
minimum radius of the circle, so that the normal force exerted on the pilot by the seat never exceeds
twice her weight ? [CutnellP5.36sim]
a. Develop the expression for the radius in
terms of v and g only.
b. Compute the radius in meters and miles. [Recall that 1m = 3.28 ft and 1 mi = 5280 ft]
Ex 2: The condition of apparent weightlessness for the passengers can be created for a brief instant
when a plane flies over the top of a vertical circle. At a speed of 215 m/s, what is the radius of
the vertical circle that a pilot must use ? [CutnellP5.37]
Ex 3: In a 1901 circus performance, Allo “Dare Devil” Diavolo introduced the stunt of riding a bicycle in
a loop-the-loop. Assuming the loop is a circle with radius R = 2.7 m, what is the least speed vtop
“Al” could have at the top of the loop if he is to remain in contact with the track ? [Halliday6.8]
Ex 4: A computer is reading data from a rotating CD-ROM. At a point that is 0.03 m from the center of
the disc, the centripetal acceleration is 120 m/s2. [CutnellP5.51mod]
a. Develop an expression for the ratio (a2 to a1) for the acceleration at another point on the disc in
terms of r1 and r2 only ?
b. What is the centripetal acceleration at a point that is 0.05 m from the center of the disc ?
AP Physics “B”
Mr. Mirro
Date: ________
Vertical Circular Motion II
Ex 1: A 2100 kg demolition ball swings at the end of a 15 m cable on the arc of a vertical circle. At the
lowest point of the swing, the ball is moving at a speed of 7.6 m/s. Determine the tension in the
cable at that (low) point. [CutnellP5.49sim]
Ex 2: The maximum tension that a 0.5 m string can tolerate is 14 N. A .25 kg ball attached to this string is
being whirled in a vertical circle. What is the maximum speed that the ball can have:
[Cutnell4EP5.39]
a. At the top of the circle ?
b. At the bottom of the circle ?
Ex 3: Even some seasoned roller coaster riders blanch (go pale) at the thought of riding the gravitron
(rotor), which is essentially a large hollow cylinder that is rotated rapidly around its central axis.
[Halliday6.12]
Before the ride begins, a rider enters the cylinder through a door on the side and stands on a floor against a
canvas-covered wall. The door is closed, and as the cylinder begins to turn, the rider, wall and floor move in
unison. When the rider’s speed reaches some predetermined value, the floor abruptly and alarming falls away.
The rider does not fall with it but instead is pinned to the wall while the cylinder rotates, as if an unseen agent
is pressing the body to the wall. Later the floor is eased back to the rider’s feet, the cylinder slows, and the
rider sinks a few centimeters to regain footing on the floor. Some riders consider this to be FUN ??
Suppose that the coefficient of static friction μs between the rider’s clothing and the canvas is 0.40
and that the cylinder’s radius is 2.1 m.
a. What minimum speed v must the cylinder and rider have if the rider is not to fall when
the floor drops ?
b. If the rider’s mass is 49 kg, what is the magnitude of the centripetal force on them ?
AP Physics
Mr. Mirro
A ball of mass M attached to a string of length L moves in a circle in a
vertical plane as shown. At the top of the circular path, the tension in the
string is twice the weight of the ball. At the bottom, the ball just clears the
ground. Air resistance is negligible. Express all answers in terms of M, L,
and g.
a. Determine the magnitude and direction of the net force on the ball
when it is at the top.
b. Determine the speed vo of the ball at the top.
The string is then cut when the ball is at the top.
c. Determine the time it takes the ball to reach the ground.
d. Determine the horizontal distance the ball travels before hitting the ground.
AP1984#1