Name: Chapter 8 Rotational Motion and Dynamics Reading Notes Section 8-1: Angular quantities A circle can be split into pieces called degrees. There are 360 degrees in a circle. A circle can be split into pieces called radians. There are about 6.8 (exactly ) radians in a circle. Example: Two coins are on the same rotating turntable. Coin A is 10 cm from the center, and coin B is 0 cm from the center. Which coin travels the fastest speed v (length per time)? Why? Which coin has the greater angular speed (angle per time)? Why? Gears Activity: There are three gears: the big gear (radius 15 cm), the medium gear (radius 10 cm), and the small gear (radius 5 cm). Gears work when their teeth interlock so that both gears rotate at the same time. For each part of the data table, interlock the two gears listed under the Gear Radius column. One of the columns tells you how much angle to rotate one gear. Your task is to say how much angle the other gear rotated. First Gear Radius First Gear Rotation Angle Second Gear Radius Second Gear Rot n Angle 5 cm 90 o 10 cm 5 cm radians 10 cm 5 cm 10 cm 135 o 5 cm 10 cm radians 5 cm 6 rotations 10 cm 5 cm 15 cm 30 o 5 cm radians 15 cm 5 cm 70 o 15 cm 5 cm 15 cm /3 radians 5 cm 6 rotations 15 cm
10 cm 60 o 15 cm 10 cm 15 cm 80 o 10 cm radians 15 cm 10 cm 40 o 15 cm 10 cm 15 cm radians 10 cm radians 15 cm 10 cm 15 cm rotations 10 cm 360 o 15 cm 10 cm 15 cm 360 o 10 cm 15 cm radians What conclusion can you draw about the radius of a gear and the angle it turns through? Use a type of relationship to help you write your conclusion. Suppose that two gears rotate at a uniform rate. A large gear has radius r 1 and angular velocity 1, and the small gear has radius r and angular velocity. Write an equation that relates these four quantities. What is the SAME for both of the gears that are turning with their teeth interlocked? Example: Two students A and B run a race on a running track. Both start at the points shown. Is this a fair race? If not, explain what could be done to make the race fair.
Section 8- & 8-3: Constant Angular acceleration & Rolling motion Write the equation that relates each linear quantity to its rotational analogue quantity. When an object rotates an angle radians, the object travels a length meters. When an object s rate of rotation is radians per second, the object s speed is v meters per second. When an object s rate of rotation increases by rad/sec every second, the object s speed increases a T m/s every second. Equation to relate these: Equation to relate these: Equation to relate these: Translational vs. Rotational Kinematics Equations TRANSLATIONAL ROTATIONAL Constant Angular Constant Velocity x vt x0 Velocity Constant Acceleration Section 8-4: Torque x v at v 0 1 at v x 0t 0 v ax v0 Constant Angular Acceleration
Walk The Plank Example: Consider a wooden plank set so that part of its length hangs over the edge of a building as shown. If a person walks out onto the plank, the plank may tip over and the person will fall. You will model this situation using a meterstick in place of the plank, and a mass in place of the person. Mass of the Plank : Mass of the Person : Pivot point of the Plank : m On the diagram, draw the forces acting on the plank for the instant when the plank barely begins to tip over. Also label the plank s axis of rotation. Write a balanced-torques equation for the instant that the plank barely begins to tip over. Your equation should include a variable that represents where the person is when the plank begins to tip. Solve your equation to find the location where the person should stand in order to begin to tip over the plank. Test your prediction with the given meterstick and mass. Bridge Support Example: Consider a bridge with two supports. An object rests on the bridge somewhere that is off-center. You are to determine the force that each support exerts on the bridge. You will model this situation with a meter-long piece of wood and a known mass. Mass of the Bridge : Mass of the Object : Object is m from left end On the diagram, draw the forces acting on the bridge. Let F 1 and F be the force of the supports. Choose one of the supports to be the axis of rotation, and then write a balanced-torques equation. Also write a second equation which is a balanced-forces equation. Solve your equations to find the force of each support F 1 and F. Use two scales to test your prediction.
Balancing Act Activity: Answer each question so that the system shown will balance. Example 1: What mass should be placed at the? so that this system balances? (Write a balanced torque equation and solve for the unknown mass). Example : What is the net torque of this system, including direction? Where should a 0.0 kg mass be placed to balance this system? Example 3: What mass should be placed at the? so that this system balances? (Write a balanced torque equation and solve for the unknown mass). Example 4: What is the net torque of this system, including direction? Where should a 0.15 kg mass be placed to balance this system?
Section 8-5: Rotational Inertia Let us define mass conceptually as a measure of. Then rotational inertia would measure. The rotational inertia of an object depends on. The basic equation for rotational inertia is, where k is called the and depends only on. The units for rotational inertia are. Newton s Second Law for Rotational Motion is. Some Moments of Inertia: Falling Meterstick Activity: Point-sized object a distance r from its axis. Solid disk of radius r (mass is uniformly distributed around the disk). Axis goes through the center. 1 Rod of length, where the axis is through the center. 1 1 Hoop of radius r (all of the mass is located at the distance r). Axis goes though the center. Sphere of radius r (mass is uniformly distributed). Axis goes though the center. 5 Rod of length, where the axis is through one end. 1 3 A meterstick is set so that 99 cm of its length is over the edge of a table as shown. The meterstick is released from rest initially horizontal. The meterstick s mass is M and its length is L. The meterstick s rotational inertia is ML /3 if it is pivoted about the table. What is the net torque acting on the meterstick in terms of M and g? Draw forces on the above diagram to help. What is the angular acceleration of the meterstick in terms of M, g, and L? At what point on the meterstick is the linear acceleration equal to g?