1. Ch 7 - Circular Motion
Circular motion: Objects moving in a circular path.

2. Measuring Rotational Motion
Rotational Motion – when an object turns about an internal axis.
Ex. Earth’s is every 24 hrs.
Axis of rotation – the line about which the rotation occurs
Arc length – the distance (s) measured along the circumference of the circle

3. Radian- an angle whose arc length is equal to its radius, which is approximately equal to 57.3°
When the arc length “s” is equal to the length of the radius, “r”, the angle θ swept by r is equal to one rad.
Any angle θ is radians if defined by:
θ = s/r

4. When a point moves 360°, θ=s/r = 2πr/r = 2π rad
Therefore, to convert from rads to degrees
θ(rad) = π θ(deg)
180°
For angular displacement,
Δθ=Δs/r
Angular displacement (in radians)= change in arc length/distance from axis

5. Example Problem
While riding on a carousel that is rotating clockwise, a child travels through an arc length of 11.5 m. If the child’s angular displacement is 165°, what is the radius of the carousel?
Ans m

6. Angular Substitutes for Linear Quantities
Linear (Straight Line)
Displacement = x
Velocity = v
Acceleration = a
Rotational
Displacement = θ
Velocity = ω
Acceleration = α

7. Angular speed – the rate at which a body rotates about an axis, expressed in radians per second
Symbol = ω (omega) Unit = rad/s
ω(avg) = Δθ/Δt
ω can also be in rev/s
To convert:
1 rev = 2π rad

8. Example Problem
A child at an ice cream parlor spins on a stool. The child turns counter-clockwise with an average angular speed of 4.0 rad/s. In what time interval will the child’s feet have an angular displacement of 8.0 rad
Ans. 6.3 s

9. Angular Acceleration – the time rate of change of angular speed, expressed in radians per second per second
avg = ω2-ω1/t2-t1 =Δω/Δt
Average angular acceleration = change in angular speed / time interval

10. Example Problem
A car’s tire rotates at an initial angular speed of 21.5 rad/s. The driver accelerates, and after 3.5 s the tire’s angular speed is 28.0 rad/s. What is the tire’s average angular acceleration during the 3.5 s time interval?
Ans. 1.9 rad/s2

11. Frequency vs. Period
Frequency – # of revolutions per unit of time. Unit: revolutions/second (rev/s).
Period – time for one revolution. Unit = second (s).
Inversely related:
t = 1/f and f = 1/t

12. Tangential Velocity Speed that moves along a circular path.
Right angles to the radii.
Direction of motion is always tangent to the circle.

13. Rotational Speed The number of rotations per unit of time.
All parts of the object rotate about their axis in the same amount of time.
Units: RPM (revolutions per minute).

14. Tangential vs. Rotational
If an object is rotating:
– All points on the object have the same rotational (angular) velocity.
– All points on the object do not have the same linear (tangential) velocity.
Tangential speed is greater on the outer edge than closer to the axis. A point on the outer edge moves a greater distance than a point at the center.
Tangential speed = radial distance x rotational speed

15. Centripetal Acceleration
The acceleration of an object moving in a circle points toward the center of the circle.
Means “center seeking” or “toward the center”.

16. 7.3 Forces that maintain circular motion

17. Consider a ball swinging on a string
Consider a ball swinging on a string. Inertia tends to make the ball stay in a straight-line path, but the string counteracts this by exerting a force on the ball that makes the ball follow a circular path.
This force is directed along the length of the string toward the center of the circle.

18. The force that maintains circular motion (formerly known as centripetal force)
Fc = (mvt2)/r
Force that maintains circular motion = mass x (tangential speed)2 ÷ distance to axis of motion
Fc = mrω2
Force that maintains circular motion = mass x distance to axis x (angular speed)2
Because this is a Force, the SI unit is the Newton (N)

19. Practice Problem
A pilot is flying a small plane at 30.0 m/s in a circular path with a radius of m. If a force of 635N is needed to maintain the pilot’s circular motion, what is the pilot’s mass?
Answer: m = 70.6 kg

20. Common Misconceptions
Inertia is often misinterpreted as a force
Think of this example: How does a washing machine remove excess water from clothes during the spin cycle?

21. Newton’s Law of Universal Gravitation
Gravitational force: a field force that always exists between two masses, regardless of the medium that separates them; the mutual force of attraction between particles of matter
Gravitational force depends on the distance between two masses

22. Newton’s Law of Universal Gravitation
Where F = Force
M1 and m2 are the masses of the two objects
R is the distance between the objects
And G = x Nm2/kg2 (constant of universal gravitation)

23. Practice Problem
Find the distance between a kg billiard ball and a kg billiard ball if the magnitude of the gravitational force is 8.92 x N.
Answer: r = 3.00 x 10-1 m