3.6.2 - Vertical Circular Motion

Essential Equations:

When describing the forces acting on an object in a vertical circle, three forces are described:

F_g - The force gravity
1. Always points downward
F_N or F_T - The normal force (when on solid surface) or force tension (when being pulled into a circle by a rope)
1. Can point in any direction
ΣF - The net force/ sum of forces
1. The Geometric sum of the F_g and F_N.

The key to Circular Motion is the sum of forces (Net Force) is equal to the centripetal force.

Steps to solving vertical circle problems:

Draw free-body diagram, identifying the directions of each force. Using common directions for each force.
Using the equation, being sure to include directions, determine the sum of the forces (ΣF).
Set the ΣF equal to F_C.
Solve for the F_N or F_T,
1. Problems will often ask what is the minium or maximum velocity that an object can move to maintain the circular path.

Horizontal to Vertical Circles

ACTIVITIES:

Moving from horizontal to vertical.
- Tension in String while moving as a pendulum.
- Identify the changes in forces as a rotating object moves from a horizontal plane to a vertical plane.
- Describe the forces at the 'top' and 'bottom' of the loop.

Objectives:

Understand that if a body moves in a circle there must be an acceleration towards the centre and therefore an unbalanced force towards the centre.
Identify the centripetal force in a variety of examples.
Looping the Loop

Horizontal v. Vertical Circles - Wireless Force Sensors - Vernier Graphical Analysis

1. How do the centripetal forces change between horizontal and vertical circles?
2. Tension at bottom of a pendulum, suppose a mass is hung from a string. The tension in the string is a constant 5-N. The mass is then pulled to the side and allowed to swing, compare the tension in the string at the bottom of the arc to the original 5-N.
3. By swinging the sensor into a full VERTICAL circle, describe how the forces change at various parts of the circle.

How does the Tension in the string change through the circle? Vertical Circle - Looping Problems:

Flipping Physics Video Resources:
Khan Academy Loop de Loop Problem
- Khan Academy - Video Intro
- Fifth Gear Episode
- Answer Part 1
- Answer Part 2
Vertical Loop Problems
- American Eagle Hills Set 2
- Problem Set 3 (Q10) - Solutions

BMX 16ft Full Pipe Loop - Red Bull Full Circle

Practice Problem: Solve on a whiteboard

The string we are using in class is called Micro-Paracord. The company claims the string can support 100-kg. Suppose you were to spin your metal ball in a perfectly vertical circle.

Your classmate wants to know the maximum velocity the ball can move (V_T) before the string breaks.

What measurements will you need to make? What units will you use to take your measurements?
Determine the velocity at which the string will break.
Predict at which point on the circle the string will break.

Suppose the string length doubled, what is the minimum velocity the metal ball needs to travel to maintain a circle?

In Class Notes and Problems - COPY

Vertical Circles

In the diagram above, the T (tension in the string) can be considered the limiting factor in the motion of the object.

At the BOTTOM of the circle the maximum tension limits the maximum velocity the object can travel, as the weight of the object and the centripetal force are added together.
At the TOP of the circle the minimum tension (T ≥ 0N) limits the minimum velocity the object can travel and maintain a circular path. If the tension is negative (mg is greater than the FC) then the object will 'fall out of the circle'.

In a rollercoaster the Tension is also considered the Normal force.

Running to the MAX....Pepsi Max (Running in a loop)

Objectives:

Introduce the quantities of circular motion, (angular velocity)
Identify the centripetal force in a variety of examples. - Looping Problems

Circular Motion FA #3

Angular Velocities

VCM, Energy, Forces and Minimal Height of Release

Suppose we would like to find the minimum height that a FRICTION-LESS roller coaster could be released from to safely pass through the loop of of the coaster, we want to define this height (H) in terms of the radius of the loop (R).

Using a combination of LCE and Forces, determine the minimum height of the coast to make it safely around the loop.

LCE - Start with a set of energy pie charts to determine the types of energy at each point.
N2L - What are the forces acting on the coaster at the top of the loop? How can these forces be used to determine the minimal velocity of the coaster?
Combine the equations to solve for the minimal height of the coaster (H) in terms of the radius of the loop (R). Express your answer as a fraction.
Suppose a the loop of a coaster were 6m. Determine the minimal height of release of the coaster.

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