Rotation about a fixed axis Rotational Motion Rotation about a fixed axis

Rotational Motion Translation Rotation Rolling

Describing Angular Motion Angular Kinematics Describing Angular Motion

Angular Position - q Displacement is now defined as angle of rotation. Polar coordinates (r, q) will make it easier to notate this motion. Dq

Dq = q1 – q0 Angular Position - q For a circle: 2p=circumference/radius So: q = s/r or s = qr s => arc length r => radius q => angle, measured in radians Remember: 360º = 2p radians Displacement: Dq = q1 – q0

Angular Velocity - w Average angular velocity Instantaneous angular velocity

Angular Direction Right-hand coordinate system w is positive when q is increasing, which happens in the counter-clockwise direction w is negative when q is decreasing, which happens in the clockwise direction

Angular Acceleration - a Average Angular Acceleration Instantaneous Angular Acceleration

At what time(s) does w = 0 rad/s? example 1 q = (t3 - 27t + 4)rad w = ? a = ? At what time(s) does w = 0 rad/s?

At t = 0s: w = 5 rad/s and q = 2 rad. example 2 a = (5t3 – 4t)rad/s2 At t = 0s: w = 5 rad/s and q = 2 rad. w = ? q = ?

Linear to Angular Conversions Position: s = qr Velocity: v = wr Acceleration: a = ar Remember, also, centripetal accleration: ac = v2/r =w2r

Angular Kinematics *constant a

example 3 A turntable starts from rest and begins rotating in a clockwise direction. 10 seconds later, it is rotating at 33.3 revolutions per minute. What is the final angular velocity (rad/s)? What is the average angular acceleration? How far did a point 10 cm from the center travel in that 10 seconds, both angularly and linearly?

example 4 In an astronaut training centrifuge (r = 15m): What constant w would give 11g’s? How fast is this in terms of linear speed? What is the translational acceleration to get to this speed from rest in 2 minutes?

Moment of Inertia “Rotational Mass”

Moment of Inertia Description of the distribution of mass Measure of an object’s ability to resist a change in rotation Note: r is the perpendicular distance from the particle to the axis of rotation.

example Two children (m=40 kg) are on the teacup ride at King’s Dominion. The childrens’ center of mass is 0.75 m from the middle of the cup. What is the moment of inertia for children in the teacup?

Moment of Inertia for Continuous Mass Distribution

Calculate I for a hoop A uniform hoop: mass (M); radius (R)

Calculate I for a thin rod A uniform thin rod: mass (M); length (L); rotating about its center of mass.

Common Moments of Inertia

Parallel Axis Theorem To calculate the moment of inertia around any axis…. Where d is the distance between the center of mass and the axis of rotation

Calculate I for a thin rod rotating around the end   A uniform thin rod: mass (M); length (L); rotating about one end.

The kinetic energy an object has due to its rotational velocity. Rotational Energy The kinetic energy an object has due to its rotational velocity.

Rotational Kinetic Energy  

example If you roll a disk and a hoop (of the same mass) down a ramp, which will win?  

The tendency of a force to cause angular motion Torque The tendency of a force to cause angular motion

Torque Torque is dependent on the amount and location of the force applied to an object. Where r is the distance between the pivot point and the force and q is the angle between r and F.

example 5 A one piece cylinder has a core section that protrudes from a larger drum. A rope wrapped around the large drum of radius, R, exerts a force, F1, to the right, while a rope wrapped around the core, radius r, exerts a force, F2 downward.. Calculate the net torque, in variables. If F1=5 N, R = 1 m, F2=6 N, and r = 0.5 m, calculate the net torque.

The “other” vector multiplication Cross Product The “other” vector multiplication

Cross product Results in a vector quantity Calculates the perpendicular product of two vectors The product of any two parallel vectors will always be zero.

The Right Hand Rule Rotational Direction is defined by the axis that it rotates around. Point your fingers along the radius Curl them in the direction of the force Your thumb will be pointing in the direction of the rotation. http://hyperphysics.phy-astr.gsu.edu/hbase/tord.html

Cross Product

Calculating Cross Product

Finding the Determinant

example 6 Find the cross product of 7i + 5j – 3k and 2i-8j + 7k. What is the angle between these two vectors?

example 7 A plumber slips a piece of scrap pipe over his wrench handle to help loosing a pipe fitting. He then applies his full weight (900 N) to the end of the pipe by standing on it. The distance from the fitting to his foot is 0.8 m, and the wrench and pipe make a 19º angle with the ground. Find the magnitude and direction of the torque being applied.

example 7 continued… r = 0.8m q = 19º F = 900 N

example 8 One force acting on a machine part is F = (-5i + 4j)N. The vector from the origin to the point applied is r = (-0.5i + 0.2j)m. Sketch r and F with respect to the origin Determine the direction of the force with the right hand rule. Calculate the torque produced by this force. Verify that your direction agrees with your calculation.

Equilibrium

Conditions for Equilibrium Net force in all directions equals zero Net torque about any point equals zero

example 9 A 2 kg seesaw has one child (30 kg) sitting 2.5 from the pivot. At what distance from the pivot should a 25 kg child sit to balance the seesaw?

example 10 Find Tcable.

example 11 Find T1 and T2.

example 12 Find m.

When the sum of the torques does not equal zero Newton’s Second Law When the sum of the torques does not equal zero

Newton’s Second Law

example 13 a = ? Fpivot L Fnormal Fgravity

example 14 FT Fg Pulley – M1, R Block – M2 Find a and FT

Angular Momentum

For a particle Remember linear momentum: So angular momentum:

Angular momentum and Torque

For a system…

To check…

Conservation If net torque is zero, then the angular momentum is constant.

Conservation For rotation about a fixed axis, we can say:

example A star with a radius of 10,000 km rotates about its axis with a period of 30 days. If it undergoes a supernova explosion and collapses into a neutron star with a radius of 3 km, what is its new period?