Rotational Motion

Rotational Motion Reading: pp. 194 – 203 (sections 8.1 – 8.3) HW #1 p. 217, question #1 p. 219, problem #1, 4, 5, 6, 15, 16, 17, 19

I. Introduction: A rotating object is one that spins on a fixed axis. The position and direction of the rotation axis will remain constant. The position of some part of the object can be specified with standard cartesian coordinates, (x,y). All objects will be assumed to rotate in a circular path of constant radius. y Both coordinates, (x, y), change over time as the object rotates. (x, y) R The object’s position can also be specified with polar coordinates, (r, q). q x For this last coordinate system, only the angle changes. The radius stays constant.

II. Definitions: Since the polar coordinates only have one changing variable, the angle, we will use this to simplify analysis of motion. position A. The ___________ is defined as where an object is in space. Here, we only need to specify the angle of the object with respect to some origin or reference line. The position of the object is measured by an angle, q, measured counterclockwise from the positive x – axis. The angle will be measured in units of ___________ rather than ___________ . radians q degrees

Radians are defined as the ratio of the length along the arc of a circle to the position of an object divided by the radius of the circle. R s q s = length along the arc measured counterclockwise from the +x – axis. R = radius of the circular path. Since q is the ratio of two lengths, the angle measurement really does not have any units. The term “radians” is just used to specify how the angle is measured. 1 revolution = 360 degrees = 2p radians

displacement B. The _____________ of the object is just the difference in its position. We define the ____________________ of the object as the difference in its angular position. angular displacement A positive Dq shows a counterclockwise {ccw} rotation, while a negative Dq shows a clockwise {cw} rotation. velocity C. Motion can also be measured through a rate of rotation, a __________. angular velocity The ____________________ of an object is defined as the amount of rotation of an object per time. The angular velocity is represented by the greek letter w (lower case omega). average angular velocity: t = elapsed time.

D. Another measure of motion is the rate of change of velocity, called an ________________. acceleration angular acceleration The ____________________ of an object is defined as the amount of change of the angular velocity of an object per time. The angular acceleration is represented by the greek letter a (lower case alpha). average angular acceleration: In general, the motion may be complex, but we will again look at constant angular acceleration cases, exactly the same way as we did back in Ch. 2. The same equations of motion can be derived for circular motion.

Ch. 8 Ch. 2 Linear Motion Rotational Motion This is only true for constant accelerations. Problem solving is the same as before.

E. The motion of an object around a circle can also be represented as actual distances along the circle and speeds tangent to the circle. vt R tangential speed The _________________ , vt, of an object is defined as the angular velocity times the radius of the circle. The tangential speed measures the actual speed of the object as it travels around the circle. length along arc average vt = elapsed time

tangential acceleration The _________________________ , at, of an object is defined as the angular acceleration times the radius of the circle. The tangential acceleration measures how the tangential speed increases or decreases over time.

Example #1: A disk rotates from rest to an angular speed of 78.00 rpm in a time of 1.300 seconds. a. What is the angular acceleration of the disk? wo = 0, w = 78.00 revolutions per minute, t = 1.300 seconds. a = ?

b. Through what angle does the disk turn?

c. Through what angle will the disk turn if it were to maintain the same angular acceleration up to 254.0 rad/s?

d. The disk has a diameter of 12.00 inches. What is the tangential speed and acceleration at the edge of the disk the moment the disk reaches 78.00 rpm?

Centripetal Force HW: See Schedule

III. Centripetal Acceleration and Force. When an object moves in a circle, the direction of its velocity is always changing. This means the object is always accelerating! For an object rotating at a constant angular speed, the acceleration of the mass is always towards the center of the motion. This kind of acceleration is called a ___________ {“center seeking”} acceleration, and is represented as ac. The amount of acceleration depends on the radius of the circular path and the speed around the circle. centripetal w = angular speed, v = tangential speed, r = radius of circular path.

Example #2: A wheel of a car has a diameter of 32.0 inches. A rock is wedged into the grooves of the tire. a. What is the centripetal acceleration on the rock if the wheel turns a rate equal to 70.0 mph?

b. What is the rotation rate of the tire, in rpm?

If there is a centripetal acceleration making an object move in a circle, then there must be an unbalanced force creating this acceleration. centripetal This force is called the _____________ force, and it also points towards the center of the circular motion. This force must be made by real forces acting on an object. The centripetal force will be the sum of the radial components of the forces acting on the object. A radial component points towards the ___________ of the circle. center

Example #3: A 0.475 kg mass is tied to the end of a 0.750 meter string and the mass is spun in a horizontal circle. If the mass makes 22.0 revolutions in a time of 2.50 seconds, what is the tension in the string holding the mass to the circular motion? rotation R T m

Centripetal Force HW #2: See Schedule Last Chance for Make-up Tests! Wednesday after school!

Example #4: A penny sits at the edge of a 12.00 inch diameter record. If the coefficient of static friction is 0.222 between the penny and the record, what is the maximum rotation rate of the record that will allow the penny to remain on the record? n rotation vertical forces balance. Fs by definition friction is: mg

set the friction force equal to the centripetal force

Example #5: A mass m on a frictionless table is attached to a hanging mass M by a cord through a hole in the table. Find the speed which m must move in order for M to stay at rest. Evaluate the speed for m = 2.00 kg, M = 15.0 kg, and r = 0.863 m. Since M is at rest, the tension force lifting it is equal to the weight of M: This tension is also the centripetal force on the mass m, causing it to spin in a circular path:

Set the two equations equal to one another:

Example #6: A common amusement park ride involves a spinning cylinder with a floor that drops away. When a high enough rotation speed is achieved, the people in the ride will stay on the side of the wall. A static friction force holds each person up. Solve for the rotation rate of the room, given the coefficient of friction for the wall and the radius of the room.

Example #6: A common amusement park ride involves a spinning cylinder with a floor that drops away. When a high enough rotation speed is achieved, the people in the ride will stay on the side of the wall. A static friction force holds each person up. Solve for the rotation rate of the room, given the coefficient of friction for the wall and the radius of the room. set the friction force equal to the weight as the vertical forces balance. Fs forces balance n m definition of force of static friction Horizontal direction: set the normal force equal to the centripetal force. mg

Combine all the information to solve for the rotation rate, w. What would be the rotation rate for a room 3.00 m wide and a carpeted wall with a coefficient of friction of 0.750?

Example #7: (Banking Angle) Determine the angle of the roadway necessary for a car to travel around the curve without relying on friction. Assume the speed of the car and the radius of the curve are given. component of the normal force that is vertical component of the normal force that is horizontal, also becomes the centripetal force

balance the vertical forces: set the net horizontal force equal to the centripetal force, with towards the center of the circular path as the positive direction: substitute in:

Example #8: Conical Pendulum. A mass of m = 1.5 kg is tied to the end of a cord whose length is L = 1.7 m. The mass whirls around a horizontal circle at a constant speed v. The cord makes an angle q = 36.9o. As the bob swings around in a circle, the cord sweeps out the surface of a cone. Find the speed v and the period of rotation T of the pendulum bob. note: divide…

simplify…

the period, T, is the time for one revolution:

Example #9: Another common amusement park ride is a rollercoaster with a loop. Determine the minimum speed at the top of the loop needed to pass through the top of the loop. There are two forces acting on the car: the weight pulling straight downwards and the normal force pushing perpendicular to the track. As the car goes through the loop, the normal force always points towards the center of the loop. The vector sum of the normal force and the radial component of the weight equals the centripetal force.

The net force for the mass at the top of the loop is: m n The faster the car goes, the greater the normal force to push the car into a circular path. The minimum speed for the car at the top of the loop is where the normal force goes to zero. mg r

Example #9: (b) Use energy conservation to find the speed of the mass at the bottom of the loop. Set the two energies equal: Divide by m and multiply by 2:

Example #9: (c) What is the necessary starting height of the mass if sliding from rest down the ramp? Set the two energies equal: Divide both sides by m and g:

AP Summary Problems! Example #10: {p. 135, problem #93} A circular curve of radius R in a new highway is designed so that a car traveling at speed vo can negotiate the turn safely on glare ice (zero friction). If the car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, then it will slip away from the center of the circle. If the coefficient of static friction is increased (from zero), a car can stay on the road while traveling at any speed within a range from vmin to vmax. Derive formulas for vmin and vmax as functions of ms, vo, g and R.

Start with the too fast scenario. The car will want to slide up the ramp, so friction will point to the bottom of the ramp. Find horizontal and vertical components of the forces: Vertical components of the forces balance: Definition of friction:

Horizontal components of the forces make the centripetal force. Take towards the left as the positive direction:

Next is the too slow scenario. The car will want to slide down the ramp, so friction will point to the top of the ramp. Find horizontal and vertical components of the forces: Vertical components of the forces balance: Definition of friction:

Horizontal components of the forces make the centripetal force. Take towards the left as the positive direction:

Example #11: A mass is tied to the end of a string and spun in a vertical circle. What is the difference in tension in the string between top and bottom of the circle? energy conservation:

subtract: now use energy conservation to eliminate speeds

Centripetal Force HW check tomorrow!

Example #12: Sonam is seated on the top of a frictionless hemispherical mound of ice of radius R. Why she is there, I do not know. A small breeze upsets the equilibrium and she starts sliding down the ice. At what vertical height from the ground does she leave the ice surface? Give the answer in terms of R. When Sonam is at the side, as shown, the sum of the radial components gives the centripetal force: h = R cos q Sonam leaves the surface when the normal force becomes zero.

Now use energy conservation to find the speed of Sonam if she starts at rest at the top. From the previous slide:

Rotational Motion