Circular Motion

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I. Intro to Circular Motion

Video Instruction Part 1

a) When an object is moving in a circle, at a speed, it is because it is _________________ __________________

b) The direction of the velocity is ____________________ to the circle

Ex 1) In which direction will the car go when it hits the ice?

c) The direction of the force and acceleration vector: __________________________

$$a_{c} = \frac{v^{2}}{r}$$

$a/r$ relationship? | $a/v$ relationship?

Ex 2) Draw the velocity, acceleration and force vector arrows for all $4.0$ positions

F = ma = __________________

Ex 3) A car is moving with a constant velocity in a circle with a diameter of $d$. It takes the car $36.0 \, \text{seconds}$ to make $6.0$ trips around the circle. Find the car's acceleration.

Ex 4) Calculate the speed an earth satellite must have to maintain a circular orbit at an altitude of $2.0 \times 10^{5} \, \text{m}$ ($200,000.0 \, \text{m}$) where the acceleration due to gravity is $9.2 \, \text{m/s}^{2}$ (radius of the earth is $6.4 \times 10^{6} \, \text{m}$)

Teacher CER Key: Fundamentals

Missing Words: (a) constant | accelerating | changing direction. (b) tangent. (c) toward the center. Relationships: Inverse ($1/r$) | Square ($v^{2}$). Formula: $F_{c} = \frac{mv^{2}}{r}$.

Ex 3 Logic: $T = 36.0 \, \text{s} / 6.0 = 6.0 \, \text{s}$. $r = d / 2.0$. $v = \frac{\pi d}{T} = \frac{\pi d}{6.0 \, \text{s}}$. $a_{c} = \frac{(\pi d / 6.0 \, \text{s})^{2}}{d / 2.0} = \frac{\pi^{2} d}{18.0 \, \text{s}^{2}}$.

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II. Vertical Circular Motion

Vertical Circles Instruction

Ex 5) Circular Motion Horizontal Surface w/ Friction: A penny with a mass $M$ is placed at a distance $R$ from the center of a moving record with a coefficient of static friction of ($\mu_{s}$). The radius of the record is $R$ and it moves with a speed of $V$.

a) What is the direction of the force on the penny?

b) What is the speed of the penny in terms of the variables given?

Ex 6) Mass on a String - Vertical Circle: Mass $M$ attached to a string moves at a constant speed of $v$ in a vertical circle of radius $r$. Find string tension $T_{top}$ and $T_{bottom}$.

a) Tension Top - Make center of circle + (Free Body Diagram)

b) Tension at Bottom

Ex 7) What is the minimum speed needed to maintain circular motion? What is $T_{top}$ at the minimum speed?

Ex 8) Roller Coaster: Find Normal Force at top for a car of mass $M$, radius $r$, speed $v$.

What minimum velocity is required to prevent passengers from falling off? Minimum $v$ occurs when ________________

Teacher CER Key: Vertical Analysis

Ex 6 Equations: Top: $T + Mg = \frac{Mv^{2}}{r}$. Bottom: $T - Mg = \frac{Mv^{2}}{r}$.

Ex 7 Logic: Critical speed is when $T_{top} = 0.0 \, \text{N}$.
$Mg = \frac{Mv^{2}}{r} \implies v = \sqrt{gr}$.

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III. Centripetal Force Problems

Solutions 1 - 3 Solutions 4 - 6

1. Find the force needed to keep a $0.50 \, \text{kg}$ rat circling $15.0$ times every $10.0 \, \text{seconds}$, if it is spun in a circle with a radius of $70.0 \, \text{cm}$ ($0.70 \, \text{m}$).

2. Find the force of friction needed to keep a $3000.0 \, \text{kg}$ car traveling on an exit ramp of $100.0 \, \text{m}$ radius, with a speed of $22.0 \, \text{m/s}$.

3. Superman Ride: A passenger on the Superman Ride of Steel is cresting a hill at $15.0 \, \text{m/s}$. If the hill has a radius of $30.0 \, \text{m}$, what force will the $50.0 \, \text{kg}$ passenger feel from his seat?

4. A certain loop coaster has a vertical loop with a $40.0 \, \text{m}$ radius. What speed at the top of the loop will make a $60.0 \, \text{kg}$ rider feel “weightless?”

5. Rotor Ride: The Rotor ride at Six Flags is a vertical cylinder that spins at $56.0 \, \text{rpm}$ ($d = 6.0 \, \text{m}$). What force does a $90.0 \, \text{kg}$ rider feel from the walls?

6. Pilot: How many g’s does a pilot feel when executing a $100.0 \, \text{m}$ radius turn at $100.0 \, \text{km/hr}$? ($100.0 \, \text{km/hr} = 27.8 \, \text{m/s}$)

Teacher CER Key: Calculations

1. Rat: $v = \frac{15.0 \cdot 2 \pi (0.70 \, \text{m})}{10.0 \, \text{s}} = 6.60 \, \text{m/s}$. $F_{c} = \frac{(0.50 \, \text{kg}) \cdot (6.60 \, \text{m/s})^{2}}{0.70 \, \text{m}} = 31.1 \, \text{N}$.

3. Superman: $F_{net} = mg - F_{N} = \frac{mv^{2}}{r} \implies (500.0 \, \text{N}) - F_{N} = 375.0 \, \text{N} \implies F_{N} = 125.0 \, \text{N}$.

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IV. Conical Pendulum

Conical Pendulum Video

Ex) $1.00\text{-meter}$ pendulum, mass $m$ moves in a circle where the string makes a radius of $0.50 \, \text{m}$.

1) Draw the FBD for the pendulum on the left side of the picture above.

2) Find the angle theta ($\theta$)

3) If tension is $T$, find $F_{net} = ma$ in y and x directions.

4) Find the velocity of the pendulum.

5) Find the period of the pendulum.

On the figure below, draw and label all the forces acting on the object when it is in the position shown in the diagram above.

Teacher CER Key: Conical

Analysis: (2) $\sin \theta = \frac{0.50}{1.00} \implies \theta = 30.0^\circ$.
(3) $T \cos 30.0^\circ = mg$ | $T \sin 30.0^\circ = \frac{mv^{2}}{r}$.
(4) $v = \sqrt{0.50 \cdot 10.0 \cdot \tan 30.0^\circ} = 1.70 \, \text{m/s}$.

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V. Concept Questions (MCQ 1-22)

MCQ Instruction Video

1. A ball fastened to string swung in vertical circle. At highest point, velocity and acceleration directions are:
A) v Left, a Down
B) v Left, a Up
C) v Right, a Down
2. Ball mass m in vertical circle. At highest point tension is:
A) mg
B) mg + ma
C) ma - mg
D) mg/ma
7. Car crests hill, driver feels weightless:
A) v = sqrt(gR)
B) v > sqrt(gR)
C) v < sqrt(gR)
11. Ball whirled in circular motion. Which is NOT true?
A) Speed constant
B) Velocity constant
C) Radius constant
13. Force F, radius r, speed v. Radius cut in half (r/2), speed becomes:
A) 2v
B) v/2
C) v/sqrt(2)
D) v*sqrt(2)
19. If speed $v$ doubles, centripetal force:
A) x2
B) x4
C) /2
22. Tension in loop-the-loop:
A) greater at bottom
B) greater at top
C) constant

MCQ Answer Key

1:A or C | 2:C | 7:A | 11:B | 13:D | 19:B | 22:A.

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VI. Free Response (FRQ 1-9)

1. Determine the minimum angle at which a frictionless road should be banked so that a car traveling at $20.0 \, \text{m/s}$ can safely negotiate the curve if the radius of the curve is $200.0 \, \text{m}$.

2. Determine the velocity that a car should have while traveling around a frictionless curve of radius $100.0 \, \text{m}$ and that is banked $20.0 \, \text{degrees}$.

3. If the road in number 2 was not frictionless, which way would the frictional force point if the car exceeded this velocity? Which way would it point if the car went slower than this velocity?

4. A curve with a radius of $50.0 \, \text{m}$ is banked at an angle of $25.0^\circ$. The coefficient of static friction between the tires and the roadway is $0.30$. (a) Find the correct speed of an automobile that does not require any friction force to prevent skidding. (b) What is the maximum speed the automobile can have before sliding up the banking? (c) What is the minimum speed the automobile can have before sliding down the banking?

5. A ball of mass $M$ attached to a string of length $L$ moves in a circle in a vertical plane. 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 $M, L, g$. (a) Determine the magnitude and direction of the net force on the ball at the top. (b) Determine the speed $v_{o}$ of the ball at the top. (c) Determine the time it takes the ball to reach the ground if the string breaks when the ball is at the top.

6. Conical Pendulum: Draw the FBD for the pendulum. Find $\theta$ if the radius is $0.50 \, \text{m}$ and the string length is $1.00 \, \text{m}$. Find the velocity and the period $T$.

7. Ball Swinger: A $0.10\text{-kg}$ ball is attached to a thread $0.80 \, \text{m}$ long and whirled in a vertical circle. The lowest point of the circle is $0.20 \, \text{m}$ above the floor. The speed of the ball at the top is $6.0 \, \text{m/s}$. (a) Determine the total mechanical energy (floor PE = 0). (b) Determine the speed at point P (lowest point). (c) Determine the tension at the bottom.

8. Lab Investigation: A $0.200\text{-kg}$ ball whirled in circle radius $0.500 \, \text{m}$. Speed $v = 3.7 \, \text{m/s}$. (1) How to determine speed. (2) Work by cord. (3) Expected tension? (4) Percent difference if actual is $5.8 \, \text{N}$.

9. Break Point: Object mass $M$ whirled with increasing speed. Breaks at $v_{o}$, radius $R$, height $h$. (a) Time required for object to hit ground after string breaks. (b) Horizontal distance from time string breaks until hit.

Teacher CER Key: FRQ

Banked 1: $\tan \theta = \frac{v^{2}}{rg} = \frac{(20.0 \, \text{m/s})^{2}}{(200.0 \, \text{m}) \cdot (10.0 \, \text{m/s}^{2})} = 0.20 \implies \theta = 11.3^\circ$.

Ball 7: (a) $E_{total} = \frac{1}{2}(0.10 \, \text{kg})(6.0 \, \text{m/s})^{2} + (0.10 \, \text{kg})(10.0 \, \text{m/s}^{2})(1.80 \, \text{m}) = 3.6 \, \text{J}$. (b) $v_{P} = \sqrt{2(3.6 \, \text{J} - 0.2 \, \text{J})/0.1 \, \text{kg}} = 8.25 \, \text{m/s}$.

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