Physics 1A Lecture 10B
Review of Last Lecture Rotational motion is independent of translational motion A free object rotates around its center of mass Objects can rotate around different axes Natural unit for angle is the radian which has dimensions of nothing (dimensionless) The polar coordinate system is the natural one to use in rotation problems Change in angle over time is the angular speed/rate Change in angular speed over time is the angular acceleration The equations of rotational kinematics are identical to those of 1D translational kinematics (although you need to know the rotation axis and the number of revolutions)
Today s Lecture Tangential motion Rotational vectors (direction of rotation) Cross product Torque
1D Linear vs. Rotational Kinematics Linear position Rotation velocity acceleration These are exactly analogous equations => rotational kinematics are identical to 1D linear kinematics
Kinematic equations for constant acceleration Linear Rotation These are exactly analogous equations => rotational kinematics are identical to 1D linear kinematics
Tangential Motion by geometry: Δθ r B A s = arc length
Tangential Motion by geometry: Δθ r B s A v tan
Tangential Motion by geometry: Δθ r B s A a tan
Tangential Motion a tan by geometry: Δθ r B s A
Angular vs. Tangential Motion Angular Motion radius = Tangential Motion
Rotation and Centripetal Acceleration Definition of centripetal acceleration: Δθ r a c B Δs A v tan centripetal acceleration goes as the square of angular speed and is proportional to radius
Rotation and Centripetal Acceleration Definition of centripetal acceleration: Δθ r a c B Δs A a tan v tan Total acceleration: (object on circular path)
What is the direction of rotation? Clockwise: in same direction as clock hands move Counterclockwise: opposite direction This is a convention a universally agreed-upon definition
Vector Quantities in Rotational Motion: The Right-hand Rule Axis of rotation The angular velocity vector points along the axis of rotation
Vector Quantities in Rotational Motion: The Right-hand Rule Axis of rotation The angular velocity vector points along the axis of rotation
Vector Quantities in Rotational Motion: The Right-hand Rule into the screen directional vector The angular velocity vector points along the axis of rotation
Vector Quantities in Rotational Motion: The Right-hand Rule out of the screen directional vector The angular velocity vector points along the axis of rotation
Torque Question: how do we speed up a bicycle wheel? Must apply a tangential force: perpendicular to position vector to interaction point
Torque Question: how do we speed up a bicycle wheel? This produces less angular acceleration Must apply a tangential force: perpendicular to position vector to interaction point
Torque Question: how do we speed up a bicycle wheel? This produces no angular acceleration Must apply a tangential force: perpendicular to position vector to interaction point
Torque Question: how do we speed up a bicycle wheel? Acceleration of wheel is less if force applied closer to center of wheel, or less force applied overall
Torque Question: how do we speed up a bicycle wheel? f Torque Acceleration of wheel is less if force applied closer to center of wheel, or less force applied overall
Mechanical Advantage moment arm or lever arm By increasing the lever arm, it is possible to apply a large torque with a modest force
Torque Question: how do we speed up a bicycle wheel? (into the screen) Acceleration vector of wheel is perpendicular to both force vector and position vector
Cross-product 90º Magnitude: equal to length of two vectors times sine of angle between them: 90º Measures degree of orthogonality of two vectors Magnitude of C is maximum when A and B are perpendicular
Cross-product 90º 90º Measures perpendicularness of two vectors Direction: perpendicular to both A and B, determined by right hand rule Order matters! If you reverse A and B and you get a different vector (saw this earlier with subtraction and division)
Cross-product 90º Direction: perpendicular to both A and B, determined by right hand rule 90º Order matters! If you reverse A and B and you get a different vector (saw this earlier with subtraction and division)
Why is the angular velocity vector perpendicular to the spinning object? Angular velocity and tangential velocity are related through the cross-product The way the cross-product is defined, the result is perpendicular to the crossed vectors ( ) with direction determined by the right-hand rule
Angular vs. Tangential Motion Angular Motion radius = Tangential Motion Δθ B A points into screen
Angular vs. Tangential Motion Angular Motion radius = Tangential Motion Δθ B A points into screen
Angular vs. Tangential Motion Angular Motion radius = Tangential Motion Δθ A B (speeding up) might point in or out of the screen
Angular vs. Tangential Motion Angular Motion radius = Tangential Motion Δθ (slowing down) A B might point in or out of the screen
For Next Time (FNT) Read Chapter 10 Do HW for Chapter 10