Download Fall 2013 Physics 172 – Recitation 5 Circular

Fall 2013
Physics 172 – Recitation 5
Circular Motion
Solution
Purpose: The purpose of this recitation is to give you experience working with the
momentum principle when dealing with circular motion.
Readings: Chapter 5.1-5.5
Learning Objectives:
2.3.1 Apply the momentum principle (in update form) to solve problems involving the
motion of objects.
2.4.2 Apply appropriate assumptions to use the position (or momentum) update formula
in the presence of changing momentum (or force).
5.1.1 Identify the forces acting on a system, including their origin and the objects they act
upon.
5.2.1 Solve statics problems, specifically,
• Draw a free body diagram indicating all forces acting on a system that is in
equilibrium
• Break vector forces into their (x,y) components based on information about
the angle
• Use the derivative form of the momentum principle to find one or more
unknown forces acting on a system in equilibrium
5.4.1 Draw a vector to represent the rate of change of momentum along a curving path
5.4.2 Apply the derivative form of the momentum principle using the parallel and
perpendicular components for a particle moving along a curving path to find one or
more unknown forces
5.5.1 Write down the formula for the rate of change of the magnitude of momentum
5.5.2 Write down the formula for the rate of change of the direction of momentum
5.5.3 Determine the acceleration of an object as it changes direction
5.5.4 Solve motion problems involving curving motion, with both constant and nonconstant speeds
Challenge Problem: A person of mass 70 kg is riding on a Ferris wheel whose radius is
4 m. The person’s speed is a constant 0.3 m/sec. Their location at a particular instant of
time is indicated by the dot in the figure below.
•
Determine the magnitude and direction of the rate of change of the person’s
momentum at the instant shown in the figure.
•
Determine the magnitude of the net force acting on the person at the instant
shown. Draw an arrow representing the direction of the net force acting on the
person at the instant shown with its tail on the dot representing the person at that
instant.
Determine the objects in the person’s surroundings that are interacting
significantly with them. Draw arrows representing each of the forces these
objects exert on the person.
Graphically show that these forces add as vectors to give your net force vector.
•
•
Questions to Answer:
a) If the person’s speed at the instant shown was the same 0.3 m/sec but it was
increasing instead of remaining constant, would the magnitude of the vector
representing the net force acting on the person at that instant be less than, equal to
or greater than it was before? Would its direction change? If so, qualitatively,
how would it change?
b) Suppose the Ferris wheel is in outer space with no significant gravitational field.
(Far away from all masses) What would be net force acting on the person on the
Ferris wheel when he/she is at the point shown on the diagram?
c) Suppose the Ferris wheel is in outer space with no significant gravitational field,
and the seat with the person breaks off the wheel. Sketch the path of motion in
which the person on seat will follow after the break off.
You will need to show the steps of GOAL1 on your white board. If you know how to do
the problem, you need to show the steps in your solution.
Be sure to include the following in your solution.
• Identify your system
• State which objects are interacting with the system
• State the fundamental principle(s) you use to solve the problems
• Draw an appropriate diagram labeling key features of the problem
• State the approximations and simplifying assumptions you use.
Gather Information
Ferris wheel whose radius = 4 m
Person mass = 70 kg
Person speed = 0.3 m/sec
Identify your system: Person
1
GOAL = Gather, Outline, Analyze, Learn
State which objects are interacting with the system: The Earth (Gravity) and the
ferris wheel seat (contact force).
State the fundamental principle(s) you use to solve the problems: Momentum
principle
Draw an appropriate diagram labeling key features of the problem: See above
State the approximations and simplifying assumptions you use:
Outline your Solution Path:
We need to determine the following:
• Determine the magnitude and direction of the rate of change of the person’s
momentum at the instant shown in the figure.
o We split the motion of objects travelling along circular paths into two
components, parallel and perpendicular to the object’s velocity.
o Initially, the Ferris wheel’s speed is constant, so the magnitude of the
person’s momentum is also constant.
o Therefore, the rate of change of the momentum does not have a parallel
component.
o The perpendicular (centripetal) component of the momentum rate of
v
 dp̂
change is | p |
= p , where R is the radius of curvature of the
dt
R
person’s “kissing circle” path. In this case R is the radius of the Ferris
wheel.
o Since the direction is perpendicular to the motion, the direction must point
either toward the center of the wheel or away and must point towards the
center of the circle (as a vector diagram showing the change in momentum
over a brief interval will confirm).
•
Determine the magnitude of the net force acting on the person at the instant
shown.
o We know that the net force on a system is related to the rate of change of
the momentum by the following relationship
•
. We determined
this answer in the previous part.
Determine the objects in the person’s surroundings that are interacting
significantly with them. Draw arrows representing each of the forces these
objects exert on the person. (Make a free body diagram.)
o Earth – the force of gravity will point straight down
o The seat must exert a force a force such that
.
Analyze the Problem:
• Determine the magnitude and direction of the rate of change of the person’s
momentum at the instant shown in the figure.
v
 dp̂
The perpendicular component of the momentum rate of change is | p |
=p ,
dt
R
where R is the radius of the Ferris wheel. So, in this case, its magnitude is
v
0.3m / s
p = (70kg × 0.3m / s)
≈ 1.6kg m2 / s 2 = 1.6N
R
4m
Since this is a low speed case, v<<c, γ = 1 and p=mv.
•
Determine the magnitude of the net force acting on the person at the instant
shown.




dp  dpˆ
v
dp
We know that Fnet = , so Fnet =
= p
= p = 1.6N
dt
dt
R
dt
Determine the objects in the person’s surroundings that are interacting
significantly with them. Draw arrows representing each of the forces these
€
objects exert on the€person.
o Earth – the force of gravity will point straight down
o The
a force a force that will balance the forces so that
 seat
 will exert

Fnet = Fgrav + Fseat
• Determine the objects in the person’s surroundings that are interacting
significantly with them. Draw arrows representing each of the forces these
objects exert on the person.
€
o Earth
 – the force of gravity will point straight down
| Fg |≈ mg = 70 × 9.8N = 686N
•
o The seat will exert a force a force that will balance the forces so that



Fnet = Fgrav + Fseat



Fseat = Fnet − Fgrav
Graphically the components are shown below:
Blue = Force due to Gravity, Turquoise = Force of the seat on the person, Red = net
Force.
Learn
•
•
The Force of the seat on the person is upwards with a small horizontal
component. (Question to pose: What is the source of the horizontal component of
the force due to the seat? What would happen if the seat were frictionless?)
While the person rotates at a constant speed, the net force will change with as the
direction of the momentum changes. (Question to ask: does both the magnitude
and direction of the net force change? This is a good one to probe. What does
change as the person continues around the Ferris wheel?) This will cause a change
in the force that the seat exerts on the person. The force of gravity will not
change.
Questions to Answer:
a) If the person’s speed at the instant shown was the same 0.3 m/sec but it was
increasing instead of remaining constant, would the magnitude of the vector
representing the net force acting on the person at that instant be less than, equal to
or greater than it was before? Would its direction change? If so, qualitatively,
how would it change?
Because the person’s instantaneous speed and the radius of curvature of their path
are the same as before, the perpendicular
component of the person’s rate of

change of momentum and, thus, F⊥ are the same as before.
On the other hand, because the person’s speed is increasing, there is now a
parallel component of force, F . The net force acting on the person is now




Fnet = F⊥ + F and its magnitude is, therefore, larger. Since F points upward and

to the right, Fnet will point along a line that runs to the right and passes a bit
above the center of the Ferris wheel.
b) Suppose the Ferris wheel is in outer space with no significant gravitational field.
(Far away from all masses) What would be net force acting on the person on the
Ferris wheel when he/she is at the point shown on the diagram?
This is to enforce understanding that net force must be the same since the
character of motion is the same. Ask them a provocative question – how could the
force be the same if gravity disappeared? Expect them to understand that it is the
NET force that must be the same, and that the seat force will be different in two
cases.
c) Suppose the Ferris wheel is in outer space with no significant gravitational field,
and the seat with the person breaks off the wheel. Sketch the path of motion in
which the person on seat will follow after the break off.
Just to enforce understanding that in the absence of force (net force zero) a person
will continue motion in a straight line