Motion in Circles  1 Circular Motion  2 Centripetal Force  3 Universal Gravitation and Orbital Motion

Objectives  Calculate angular speed in radians per second.  Calculate linear speed from angular speed and vice- versa.  Describe and calculate centripetal forces and accelerations.  Describe the relationship between the force of gravity and the masses and distance between objects.  Calculate the force of gravity when given masses and distance between two objects.  Describe why satellites remain in orbit around a planet.

Vocabulary  angular displacement  angular speed  axis  centrifugal force  centripetal acceleration  centripetal force  circumference  ellipse  gravitational constant  law of universal gravitation  linear speed  orbit  radian  revolve  rotate  satellite

Motion in Circles Investigation Key Question: How do we describe circular motion?

Motion in Circles  We say an object rotates about its axis when the axis is part of the moving object.  A child revolves on a merry-go-round because he is external to the merry- go-round's axis.

Motion in Circles  Earth revolves around the Sun once each year while it rotates around its north- south axis once each day.

Motion in Circles  Angular speed is the rate at which an object rotates or revolves.  There are two ways to measure angular speed  number of turns per unit of time (rotations/minute)  change in angle per unit of time (deg/sec or rad/sec)

Circular Motion  A wheel rolling along the ground has both a linear speed and an angular speed.  A point at the edge of a wheel moves one circumference in each turn of the circle.

The relationship between linear and angular speed  The circumference is the distance around a circle.  The circumference depends on the radius of the circle.

The relationship between linear and angular speed  The linear speed (v) of a point at the edge of a turning circle is the circumference divided by the time it takes to make one full turn.  The linear speed of a point on a wheel depends on the radius, r, which is the distance from the center of rotation.

The relationship between linear and angular speed C = 2 π r Radius (m) Circumference (m) v = d t Distance (m) Speed (m/sec) Time (sec) 2πr2πr

The relationship between linear and angular speed v =  r Radius (m) Linear speed (m/sec) Angular speed (rad/sec) * Angular speed is represented with a lowercase Greek omega (ω).

 You are asked for the children’s linear speeds.  You are given the angular speed of the merry-go-round and radius to each child.  Use v = ωr  Solve:  For Siv: v = (1 rad/s)(4 m) v = 4 m/s.  For Holly: v = (1 rad/s)(2 m) v = 2 m/s. Calculate linear from angular speed Two children are spinning around on a merry-go- round. Siv is standing 4 meters from the axis of rotation and Holly is standing 2 meters from the axis. Calculate each child’s linear speed when the angular speed of the merry go-round is 1 rad/sec?

The units of radians per second  One radian is the angle you get when you rotate the radius of a circle a distance on the circumference equal to the length of the radius.  One radian is approximately 57.3 degrees, so a radian is a larger unit of angle measure than a degree.

The units of radians per second  Angular speed naturally comes out in units of radians per second.  For the purpose of angular speed, the radian is a better unit for angles.  Radians are better for angular speed because a radian is a ratio of two lengths.

Angular Speed  =  t Angle turned (rad) Time taken (sec) Angular speed (rad/sec)

 You are asked for the angular speed.  You are given turns and time.  There are 2π radians in one full turn. Use: ω = θ ÷ t  Solve: ω = (6 × 2π) ÷ (2 s) = 18.8 rad/s Calculating angular speed in rad/s A bicycle wheel makes six turns in 2 seconds. What is its angular speed in radians per second?

Relating angular speed, linear speed anddisplacement  As a wheel rotates, the point touching the ground passes around its circumference.  When the wheel has turned one full rotation, it has moved forward a distance equal to its circumference.  Therefore, the linear speed of a wheel is its angular speed multiplied by its radius.

 You are asked for the angular speed in rpm.  You are given the linear speed and radius of the wheel.  Use: v = ωr, 1 rotation = 2π radians  Solve: ω = v ÷ r = (11 m/s) ÷ (0.35 m) = 31.4 rad/s.  Convert to rpm: 31.4 rad x 60 s x 1 rotation = 300 rpm 1 s 1 min 2 π rad Calculating angular speed from linear speed A bicycle has wheels that are 70 cm in diameter (35 cm radius). The bicycle is moving forward with a linear speed of 11 m/s. Assume the bicycle wheels are not slipping and calculate the angular speed of the wheels in rpm.

Motion in Circles 1 Circular Motion 2 Centripetal Force 3 Universal Gravitation and Orbital Motion

2 Centripetal Force Investigation Key Question: Why does a roller coaster stay on a track upside down on a loop?

Centripetal Force  We usually think of acceleration as a change in speed.  Because velocity includes both speed and direction, acceleration can also be a change in the direction of motion.

Centripetal Force  Any force that causes an object to move in a circle is called a centripetal force.  A centripetal force is always perpendicular to an object’s motion, toward the center of the circle.

Calculating centripetal force  The magnitude of the centripetal force needed to move an object in a circle depends on the object’s mass and speed, and on the radius of the circle.

Centripetal Force F c = mv 2 r Linear speed (m/sec) Radius of path (m) Centripetal force (N) Mass (kg)

 You are asked to find the centripetal force.  You are given the radius, mass, and linear speed.  Use: F c = mv 2 ÷ r  Solve: F c = (50 kg)(6 m/s) 2 ÷ (3 m) = 600 N Calculating centripetal force A 50-kilogram passenger on an amusement park ride stands with his back against the wall of a cylindrical room with radius of 3 m. What is the centripetal force of the wall pressing into his back when the room spins and he is moving at 6 m/sec?

Centripetal Acceleration  Acceleration is the rate at which an object’s velocity changes as the result of a force.  Centripetal acceleration is the acceleration of an object moving in a circle due to the centripetal force.

Centripetal Acceleration a c = v 2 r Speed (m/sec) Radius of path (m) Centripetal acceleration (m/sec 2 )

 You are asked for centripetal acceleration and a comparison with g (9.8 m/s 2 ).  You are given the linear speed and radius of the motion.  Use: a c = v 2 ÷ r  4. Solve: a c = (10 m/s) 2 ÷ (50 m) = 2 m/s 2  The centripetal acceleration is about 20%, or 1 / 5 that of gravity. Calculating centripetal acceleration A motorcycle drives around a bend with a 50- meter radius at 10 m/sec. Find the motor cycle’s centripetal acceleration and compare it with g, the acceleration of gravity.

Centrifugal Force  Although the centripetal force pushes you toward the center of the circular path...it seems as if there also is a force pushing you to the outside.  This “apparent” outward force is often incorrectly identified as centrifugal force.  We call an object’s tendency to resist a change in its motion its inertia.  An object moving in a circle is constantly changing its direction of motion.

Centrifugal Force  This is easy to observe by twirling a small object at the end of a string.  When the string is released, the object flies off in a straight line tangent to the circle.  Centrifugal force is not a true force exerted on your body.  It is simply your tendency to move in a straight line due to inertia.