Two-Dimensional Rotational Dynamics 8.01 W10D2 Fall, 2006

Main Idea: Fixed Axis Rotation of Rigid Body Torque produces angular acceleration about center of mass is the moment of inertial about the center of mass is the angular acceleration about center of mass

Torque as a Vector Force exerted at a point P on a rigid body. Vector from a point S to the point P. P  S Torque about point S due to the force exerted at point P:

Summary: Cross Product Magnitude: equal to the area of the parallelogram defined by the two vectors Direction: determined by the Right-Hand-Rule

Properties of Cross Products

Cross Product of Unit Vectors Unit vectors in Cartesian coordinates

Components of Cross Product

Concept Question: Cross Product Consider two vectors with x > 0 and with Fx > 0 and Fz > 0 . The cross product points in the + x-direction -x-direction +y-direction -y-direction +z-direction -z-direction None of the above directions

Recall from Last Class: Rotational Kinematics Individual element of mass Radius of orbit Tangential velocity Tangential acceleration Radial Acceleration

Dynamics: Newton’s Second Law and Torque about S Tangential force on mass element produces torque Newton’s Second Law Torque about S z-component of torque about S

Moment of Inertia and Torque Component of the total torque about an axis passing through S is the sum over all elements Recall: Moment of Inertia about and axis passing through S : Summary:

Concept Question: Chrome Inertial Wheel A fixed torque rotates is applied to the shaft of the chrome inertial wheel. If the four weights on the arms of the are slid out, the component of the angular acceleration along the shaft direction will increase. decrease. remain the same.

Demo: Chrome Inertial Wheel For a fixed torque: vary the moment of inertia will vary the angular acceleration.

Demo: Moment of Inertia Wheel Measuring the moment of inertia.

Analysis: Measuring Moment of Inertia Free body force diagrams and force equations: Rotational equation: Constraint: Solve for moment of inertia: Time to travel distance s:

Properties of Wheel Radius of disc: Mass of disc: Mass of weight holder: Theoretical result:

Group Problem: Atwood’s Machine A pulley of mass mp, radius R, and moment of inertia Icm about the center of mass, is suspended from a ceiling. An inextensible string of negligible mass is wrapped around the pulley and attached on one end to an object of mass m1 and on the other end to an object of mass m2 , with m1 > m2 . At time t = 0, the objects are released from rest. Find the magnitude of the acceleration of the objects.

Analysis: Atwood’s Machine Free body force diagrams and force equations:

Analysis: Atwood’s Machine Rotational motion equation: Force equations: Constraint condition: Acceleration: Time to travel distance d:

Experiment 05: Moment of Inertia

Apparatus Connect output of phototransistor to channel A of 750. Connect output of tachometer generator to channel B of 750. Connect power supply. Red button is pressed: Power is applied to motor. Red button is released: Rotor coasts: Read output voltage using LabVIEW program. Use black sticker or tape on white plastic rotor for generator calibration.

Calibrate Tachometer-generator Spin motor up to full speed, let it coast. Measure and plot voltages for 0.25 s period. Sample Rate: 5000 Hz. Count rotation periods to measure ω. Program calculates average output voltage, angular velocity, and the calibration factor angular velocity per volt periods.

Rotor Moment of Inertia Plot only the generator voltage for rest of experiment. Use a 55 gm weight to accelerate the rotor.

Analysis: moment of inertia Force and rotational equations while weight is descending: Constraint: Rotational equation while slowing down Solve for moment of inertia: Speeding up Slowing down

Understand graph output to measure IR The angular frequency along line A-B is increasing because the weight has hit the floor and is tension in the string is no longer applying a torque. the weight is descending and the tension in the string is applying a torque. 3. for some other reason.

Understand graph output to measure IR The slope of the line B-C is equal to the angular acceleration after the weight has hit the floor. angular acceleration before the weight has hit the floor. 3. Neither of the above.

Measure IR: results Measure and record α1 and α2. For your report, calculate IR: