Ch 7 - Circular Motion

• Circular motion: Objects moving in a circular path.

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

• 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

• When a point moves 360°,θ=s/r = 2πr/r = 2π radTherefore, to convert from rads to degrees

θ(rad) = π θ(deg)

180°For angular displacement,Δθ=Δs/rAngular displacement (in radians)= change in

arc length/distance from axis

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. 3.98 m

Angular Substitutes for Linear Quantities

• Linear (Straight Line)– Displacement = x– Velocity = v– Acceleration = a

• Rotational– Displacement = θ– Velocity = ω– Acceleration = α

• 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

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

• 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

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

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

Tangential Velocity

• Speed that moves along a circular path.

• Right angles to the radii.

• Direction of motion is always tangent to the circle.

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).

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

v r

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”. 2

c

va

r

2ca r

7.3 Forces that maintain circular motion

• 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.

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)

Practice Problem

• A pilot is flying a small plane at 30.0 m/s in a circular path with a radius of 100.0 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

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?

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

Newton’s Law of Universal Gravitation

221

r

mGmF

Where F = ForceM1 and m2 are the masses of the two objects

R is the distance between the objectsAnd G = 6.673 x 10-11 Nm2/kg2 (constant of universal gravitation)

Practice Problem

• Find the distance between a 0.300 kg billiard ball and a 0.400 kg billiard ball if the magnitude of the gravitational force is 8.92 x 10-11 N.

• Answer: r = 3.00 x 10-1 m