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ROTATIONAL POSITION & DISPLACEMENT Rotational Motion is motion around a point, that is, in a path. - The rotational equivalent of linear POSITION ( ) is Rotational/Angular position ( ). X LINEAR POSITION ROTATIONAL POSITION - How far you are from the. - Measured in. - Origin is where. - Origin is. - Direction (+/-) is. - How far you are from the. - Measured in. - Origin is where. - Origin is : - Always at the. - Direction (+/-) is : CW CCW The rotational equivalent of linear DISPLACEMENT ( ) is Rotational Displacement ( ). - These two quantities are LINKED by an equation (and r = radial distance, radius ): - This equation speaks. Input must be in radians. Output will be in radians. - One radian is approximately 57 o. To convert between radians and degrees, use: ΔX ΔX = ( ) = EXAMPLE: An object moves along a circle of radius 10 m from 30 o above the positive x-axis to 120 o above the +x-axis. Calculate the object s (a) angular displacement, and (b) linear displacement. Page 2
DISPLACEMENT IN MULTIPLE REVOLUTIONS If you make one full revolution around a circle: ΔΘ = = ΔX = =. - If you make N full revolutions: ΔΘ = = ΔX =. - To find out how many revolutions you go through, simply divide ΔΘ by or. - To find out how far from 0 o you end up after many revolutions, subtract by 360 o until Θ < 360 o (or Θ < 2 π). EXAMPLE: Starting from 0 o, you make two 2.2 revolutions around a circular path of radius 20 m. (a) What is your rotational displacement, in degrees? (b) How many degrees away from 0 o are you? (c) What is your linear displacement? Page 3
PRACTICE: ROTATIONAL DISPLACEMENT PRACTICE: While you drive, your tires, all of radius 0.40 m, rotate 10,000 times. How far did you drive, in meters? Page 4
PRACTICE: ROTATIONAL DISPLACEMENT PRACTICE: An object moves a total distance of 1,000 m around a circle of radius 30 m. How many degrees does the object go through? BONUS: How many complete revolutions does it make? Page 5
PRACTICE: ROTATIONAL DISPLACEMENT PRACTICE: A car travels a total of 2,000 m and 1140 o around a circular path, starting from 0 o. What is the radius of the circular path? BONUS: How far (in degrees) from 0 o does the car end up? Page 6
ROTATIONAL VELOCITY & ACCELERATION The rotational equivalents of linear velocity and acceleration are Rotational Velocity and Rotational Acceleration: v,avg = [ ] = [ ] a = [ ] = [ ] There are 3 additional variables that describe how quickly something rotates (similar to w). They are all related: w = = = 1 RPM = ( ) 1 Hz = - Often we will convert from any of these three back to Note that rotational equations work for both: (1) Points Masses ( ) moving in a circular path; or (2) Rigid Body/Shape ( ) rotating around themselves. EXAMPLE 1: A 30-kg disc of radius 2 m rotates at a constant 120 RPM. Calculate its (a) period, (b) angular speed. EXAMPLE 2: Calculate the rotational velocity for the Earth as it (a) rotates around itself, (b) rotates around the Sun. Page 7
PRACTICE: ROTATIONAL VELOCITY & ACCELERATION PRACTICE: Calculate the rotational velocity (in rad/s) of a clock s minute hand. EXTRA: Calculate the rotational velocity (in rad/s) of a clock s hour hand. Page 8
PRACTICE: ROTATIONAL VELOCITY & ACCELERATION PRACTICE: A wheel of radius 5 m accelerates from 60 RPM to 180 RPM in 2 s. Calculate its angular acceleration. Page 9
MOTION EQUATIONS FOR ROTATION Just like in linear motion, there are 4* equivalent MOTION equations for Rotation. Same equations, different letters. - You often use these when given a lot of rotational quantities. Same process: List variables, pick equation, solve. LINEAR EQUATIONS ROTATIONAL EQUATIONS vf = vi + a t vf 2 = vi 2 + 2 a Δx Δx = vi t + ½ a t 2 Δx = ½ (vi + vf) t * EXAMPLE 1: A wheel initially at rest accelerates around its central axis, with a constant 4 rad/s 2 until it reaches 80 rad/s. (a) By the time it reaches 80 rad/s, how many degrees will it have rotated through? (b) How long (in s) does this take? EXAMPLE 2: A very heavy disk, 20 m in radius, takes 1 hour to make a complete revolution, accelerating from rest at a constant rate. What rotational velocity will the disk have 1 hour after it starts accelerating? Page 10
PRACTICE: MOTION EQUATIONS FOR ROTATION PRACTICE: A tiny object spins with 5 rad/s around a circular path of radius 10 m. The object then accelerates at 3 rad/s 2. Calculate its angular speed 8 s after starting to accelerate. BONUS: Calculate its linear displacement in the 8 s. Page 11
PRACTICE: MOTION EQUATIONS FOR ROTATION PRACTICE: The turntable of a DJ set is spinning at a constant rate just before it is turned off. If the turntable decelerates at 3 rad/s 2 and goes through an additional 30 rotations before stopping, how fast (in RPM) was the turntable initially spinning? BONUS: How long (in seconds) does the turntable take to stop? Page 12
CONVERTING BETWEEN LINEAR AND ROTATIONAL There are tiny equations that LINK linear (aka tangential) and rotational (aka angular) variables: LINEAR ROTATIONAL LINK x Θ -- Δx = x xo ΔΘ = Θ Θo Δx = r ΔΘ v = Δx / Δt w = ΔΘ / Δt v,t = a = Δv / Δt α = Δw / Δt a,t = - There are 4 types of acceleration. The equation a,t = refers to acceleration. More soon! - When a Shape/Rigid Body rotates around itself, ALL rotational quantities ( ΔΘ, w, α ) are the same at every point. - Linear speeds (v,t = r w) may be different, since they depend on. EXAMPLE 1: A wheel of radius 8 m spins around its central axis at 10 rad/s. Find the angular AND linear speeds at a point: (i) at the middle of the wheel (on its central axis); (ii) at a distance of 4 m from the wheel s center; (iii) at the edge of the wheel. EXAMPLE 2: A small object rotates at the end of a light string. The object reaches 120 RPM from rest in just 4 seconds. If the object s tangential acceleration after the 4 seconds is 15 m/s 2, calculate the length of the string. Page 13
PRACTICE: CONVERTING BETWEEN LINEAR AND ROTATIONAL PRACTICE: A disc of radius 10 m rotates around itself with a constant 180 RPM. Calculate the linear speed at a point 7 m from the center of the disc. Page 14
PRACTICE: CONVERTING BETWEEN LINEAR AND ROTATIONAL PRACTICE: A rock rotates around a light, 4-m long string. The rock is initially at rest, but reaches 150 RPM in 3 seconds. Calculate its tangential acceleration after 3 s. BONUS: Calculate its tangential speed after 3 s. Page 15
PRACTICE: ROTATIONAL KINEMATICS PRACTICE: A 4 m long blade initially at rest begins to spin with 3 rad/s 2 around its axis, which is located at the middle of the blade. It accelerates for 10 s. Find the tangential speed of a point at the tip of the blade 10 s after it starts rotating. Page 16
TYPES OF ACCELERATION IN ROTATION There are FOUR types of acceleration in rotation problems - BUT some exist only if you re accelerating (spinning faster): - Centripetal / Radial (linear) - Tangential (linear) - Total / Acceleration (linear) - Rotational / Angular - You always have v T, a C (aka a RAD), and w: - IF accelerating: you also have a T and α: - a,c. - a,t (and α). - The equation a,t = r α is a way to remember that a,t and α are connected. If one is zero, the other has to be zero. - Note that IF a,t = 0, then a = becomes a = = a,c. EXAMPLE 1: A carousel 10 m in radius completes one cycle every 45 s. A boy stands at the edge of the carousel. Find his: (a) Tangential velocity (b) Angular acceleration (c) Radial acceleration (d) Tangential acceleration (e) Total linear acceleration EXAMPLE 2: A carousel 16 m in radius accelerates from rest with 0.05 rad/s 2. A boy stands at the edge of the carousel. After the carousel has accelerated for 10 s, calculate the boy s: (a) Tangential velocity (b) Tangential acceleration (c) Radial acceleration (d) Angular acceleration (e) Total linear acceleration Page 17
PRACTICE: ROTATIONAL KINEMATICS PRACTICE: A large disc of radius 10 m initially at rest takes 200 full revolutions to reach 30 RPM. Calculate the total linear acceleration of a point at half way between the disc s center and its edge, once the disc reaches 30 RPM. (You may assume it continues accelerating past that point) Page 18
PRACTICE: ROTATIONAL KINEMATICS PRACTICE: An object of negligible size moves in a circular path of radius 20 m with 90 RPM. Find its radial acceleration. Page 19
ROLLING MOTION (FREE WHEELS) So far we have seen Point Masses in a circular path OR Shapes/Rigid Bodies around themselves FIXED WHEEL - In some problems, Shapes/Rigid Bodies are BOTH Rotating ( ) AND MOVING ( ) FREE WHEEL - Cylinder rotating around FIXED axis: - Cylinder rotating around FREE axis: w 0 BUT v,cm 0 w 0 AND v,cm 0 - If FREE Axis, the total velocity (linear) at the center of mass (usually middle), top, and bottom of the wheel are: EXAMPLE 1: A wheel of radius 0.30 cm rolls without slipping along a flat surface with 10 m/s. Calculate (a) the angular speed of the wheel; and (b) the speed of a point at the bottom of the wheel, relative to the floor. EXAMPLE 2: When a car accelerates from rest for 10 s, its tires experience 8 rad/s 2. The tires are 0.40 m in radius. Calculate: (a) the angular speed of the tires after 10 s; (b) the speeds of points at the top, center, and bottom of the tire. Page 20
PRACTICE: ROLLING MOTION PRACTICE: A long, light rope is wrapped around a cylinder of radius 40 cm, which is at rest on a flat surface, free to move. You pull horizontally on the rope, so it unwinds at the top of the cylinder, causing it to begin to roll without slipping. You keep pulling until the cylinder reaches 10 RPM. Calculate the speed of the rope at the instant the cylinder reaches 10 RPM. Page 21
CONNECTED WHEELS (STATIC) Problems where two wheels (discs, cylinders, etc.) are connected are fairly common in Rotational Kinematics. - In some cases, the wheels rotate around a FIXED Axis (w BUT no v,cm) fixed pulleys/gears, static bicycles - In others, the wheels rotate around a FREE Axis (w AND v,cm) bicycle moving on the ground - Whenever a chain connects two wheels, we have, which yields a set of 4 related equations: EXAMPLE 1: Two gears (R1 = 2 m, R2 = 3 m) are free to rotate about fixed axes, with a light chain that runs around them, so they spin together (as above). When you give the smaller pulley 40 rad/s, what angular speed will the larger one have? EXAMPLE 2: Two pulleys, with radii 0.3 m (left) and 0.4 m (right) attached to a table as shown below. A light cable runs through the edge of both pulleys, with one end connected to a mass. You pull down on the other end, causing the pulleys to spin (and the block to move up). When the cable has speed 5 m/s, what will the angular speed of each pulley be? m Page 22
CONNECTED WHEELS: BICYCLES (STATIC) Bicycle problems are a bit more complicated than static pulleys/gears, because there are more parts (5 in total): - The pedals (1) cause the middle sprocket (2) to spin - The chain connects both sprockets - The back sprocket (3) is connected to the back wheel (4) - If the bike is NOT free to move (wheels don t touch ground), the front wheel (5) doesn t move or spin (v5 = w5 = 0). EXAMPLE 1: You turn your bicycle upside down for maintenance. The middle and back sprockets have diameters 16 cm and 10 cm, respectively. You spin the pedals at 8 rad/s. Calculate the resulting angular velocity for the: (a) middle sprocket (b) back sprocket (c) back wheel (d) front wheel EXAMPLE 2: You lift your bicycle slightly and begin to spin its back wheel. The middle and back sprockets have diameters 2D and D, respectively. If you spin the back wheel at X RPM, at how many RPM (in terms of X), will the pedals spin? Page 23
CONNECTED GEARS: MOVING BICYCLES Remember: If the bike doesn t move when the wheels spin (FIXED Axis) v,front = w,front =. - If the bike IS moving (FREE Axis: w AND v), you also have that v,cm,front = v,cm,back = v,bike - Remember: For Free Axis, we have - For most bicycles R,front = R,back EXAMPLE 1: The wheels on your bike have radius of 66 cm. If you ride with 15 m/s, calculate the: (a) linear speeds at the center of mass of both wheels; and (b) the angular speeds of both wheels. EXAMPLE 2: The wheels on your bike have radius 70 cm. The middle and back sprockets on your bike have radii 15 cm and 8 cm, respectively. If you ride with 20 m/s, calculate the angular speed of the: (a) front wheel (b) back wheel (c) back sprocket (d) middle sprocket Page 24