Checking Understanding: Circular Motion Dynamics When the ball reaches the break in the circle, which path will it follow? Slide 6-21
Answer When the ball reaches the break in the circle, which path will it follow? C. Slide 6-22
Additional Questions A coin sits on a rotating turntable. At the time shown in the figure, which arrow gives the direction of the coin s velocity? Slide 6-45
Answer: A A coin sits on a rotating turntable. At the time shown in the figure, which arrow gives the direction of the coin s velocity? A Slide 6-46
Additional Questions A coin sits on a rotating turntable. At the instant shown, suppose the frictional force disappeared. In what direction would the coin move? Slide 6-49
A coin sits on a rotating turntable. Answer At the instant shown, suppose the frictional force disappeared. In what direction would the coin move? A Slide 6-50
Checking Understanding When a ball on the end of a string is swung in a vertical circle, the ball is accelerating because A. the speed is changing. B. the direction is changing. C. the speed and the direction are changing. D. the ball is not accelerating. Slide 6-13
Answer When a ball on the end of a string is swung in a vertical circle, the ball is accelerating because A. the speed is changing. B. the direction is changing. C. the speed and the direction are changing. D. the ball is not accelerating. Slide 6-14
Checking Understanding: Circular Motion Dynamics For the ball on the end of a string moving in a vertical circle: What is the direction of the net force on the ball? A. tangent to the circle B. toward the center of the circle C. there is no net force Slide 6-19
Answer For the ball on the end of a string moving in a vertical circle: What is the direction of the net force on the ball? A. tangent to the circle B. toward the center of the circle C. there is no net force Slide 6-20
For the ball on the end of a string moving in a vertical circle: What force is producing the centripetal acceleration of the ball? A. gravity B. air resistance C. normal force D. tension in the string Checking Understanding: Circular Motion Dynamics Slide 6-17
Answer For the ball on the end of a string moving in a vertical circle: What force is producing the centripetal acceleration of the ball? A. gravity B. air resistance C. normal force D. tension in the string Slide 6-18
Additional Questions A coin sits on a rotating turntable. At the time shown in the figure, which arrow gives the direction of the frictional force on the coin? Slide 6-47
A coin sits on a rotating turntable. Answer At the time shown in the figure, which arrow gives the direction of the frictional force on the coin? D Slide 6-48
Reading Quiz For uniform circular motion, the acceleration A. is parallel to the velocity. B. is directed toward the center of the circle. C. is larger for a larger orbit at the same speed. D. is always due to gravity. E. is always negative. Slide 6-6
Answer For uniform circular motion, the acceleration A. is parallel to the velocity. B. is directed toward the center of the circle. C. is larger for a larger orbit at the same speed. D. is always due to gravity. E. is always negative. Slide 6-7
Summary Speed = v = Δd/Δt = 2πr/T at radius, r, Slide 6-41
Checking Understanding When a ball on the end of a string is swung in a vertical circle: What is the direction of the acceleration of the ball? A. Tangent to the circle, in the direction of the ball s motion B. Toward the center of the circle Slide 6-15
Answer When a ball on the end of a string is swung in a vertical circle: What is the direction of the acceleration of the ball? A. Tangent to the circle, in the direction of the ball s motion B. Toward the center of the circle Slide 6-16
Angular Displacement,, and Angular Velocity, ω is the angle between position 1 and 2. is the amount of rotation measured in degrees or radians (we ll use radians since they are the SI unit for ). 2 r 1 For an object rotating about a fixed axis, the angular displacement,, is the angle swept out by a line passing through any point on the body and intersecting the axis of rotation perpendicularly. By convention, the angular displacement is positive if it is counterclockwise.
Angular Displacement, For an object rotating, angular displacement is found by: (in radians) = Arc length = s radius = f i where i is sometimes zero. r
If determining for a complete circle, or one complete cycle: Arc length = 2 r (for one revolution) = (arc length) / r = (2 r) / r = 2 radians 1 revolution = 2 radians = 360 o So, 1 radian = 360 o / 2 = 57.3 o
UNIFORM CIRCULAR MOTION (UCM) Angular displacement = Δθ = the angle displaced, measured in radians, while an object undergoes UCM Δθ= ω Δt Angular Velocity = ω (Greek: Omega) ω = 2 π f and ω = Δθ/Δt All points on a rotating object rotate through the same angle in the same time, and have the same frequency. Angular velocity is also known as the turning rate. Angular velocity: all points on a rotating object have the same angular velocity, ω, (same turning rate) but different speeds, v, and v =ωr, depending on the location of that point. v =ωr
ω is positive if object is rotating counterclockwise. (Negative if rotation is clockwise.) Conversion: 1 revolution = 2 π rad
Angular Velocity Angular Velocity = the turning rate expressed in units of revolutions per second, or metric: radians per second. Angular velocity = angular displacement / elapsed time = / t units: radians/second or rad/s For one cycle: t = T = period = time for one cycle And, =s/r= (2 r)/r = 2 radians = angular displacement for one cycle = / t = 2 radians / T = 2 / T = 2 /T And since T=1/f, where f = frequency = 2 / (1/f) = 2 f = 2 f Angular velocity is sometimes called angular frequency!
Checking Understanding Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The angular velocity of A is twice that of B. B. The angular velocity of A equals that of B. C. The angular velocity of A is half that of B. Slide 7-13
Answer Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The angular velocity of A is twice that of B. B. The angular velocity of A equals that of B. C. The angular velocity of A is half that of B. All points on the turntable rotate through the same angle in the same time. ω = θ/ t All points have the same period, therefore, all points have the same frequency. ω = 2 π f Slide 7-14
Angular velocity,, or the turning rate is the same everywhere on the rotating body/object, b/c / t =2 /T is the same. But, instantaneous velocity, the velocity tangent to the circular path called tangential velocity, v T is greater for points (places) farther from the axis. The magnitude of the tangential velocity is referred to as the tangential speed. v=d/t = 2 r/t = 2 rf v T = r Important to realize v T v T v T V T
Checking Understanding Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The speed of A is twice that of B. B. The speed of A equals that of B. C. The speed of A is half that of B. Slide 7-15
Answer Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The speed of A is twice that of B. B. The speed of A equals that of B. C. The speed, v, of A is half that of B. v = wr Twice the radius means twice the speed, v. Therefore, B moves with twice the speed of A. Slide 7-16
Uniform Circular Motion a c = From: v T = r = 2 π r f v = v T a = a c Slide 6-23
Circular Motion And v = 2πrf = tangential speed or speed at that place (at r) a = a c