Rotational Motion : Rotational Motion How the Wheels on the Bus Go…
Dave McCallister
Feb 20, 2010
Torque : Torque Rotation
Physics
Mr. McCallister
Extended objects : Extended objects Point masses are useful to determine the motion of an object’s center of mass, but not to analyze rotation.
Objects that spin must be modeled as extended objects (objects with definite size and shape)
Torque : Torque Torque is a measure of a force’s ability to cause an object to rotate. It is symbolized by t and measured in N m.
Torque is found by multiplying the size of the force by the lever arm.
The lever arm is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force, symbolized by d and measured in m.
Formula: t = F d sin ?
Net torque : Net torque Net torque is the sum of all individual torques acting on an object.
Counterclockwise is designated positive, clockwise negative. Or: use right-hand-rule.
Stop to Think… : Stop to Think… Find the magnitude of a torque produced when a 4.0 N force is applied to a door at a perpendicular distance of 1.0 m away from the hinges.
Center of Mass, Moment of Inertia, & Rotational Equilibrium : Center of Mass, Moment of Inertia, & Rotational Equilibrium Rotation
Physics
Mr. McCallister
Center of Mass : Center of Mass Center of mass is “the point at which all the mass of the body can be considered to be concentrated when analyzing translational motion”
For simple objects of uniform density, the center of mass will be located at geometric center of the object.
However, the center of mass may not necessarily be located within the volume of the object (i.e. doughnut, boomerang).
Center of Mass : Center of Mass If gravity is the only force acting on a rotating object:
The object will rotate about its center of mass
The center of mass will follow a parabolic trajectory (projectile motion)
Moment of Inertia : Moment of Inertia Moment of inertia is “the tendency of a body rotating about a fixed axis to resist a change in rotational motion”
In other words, moment of inertia is a measure of an object’s resistance to being rotated (similar to inertia being a measure of an object’s resistance to being accelerated)
Symbol: I , Unit: kg m2
Moment of Inertia : Moment of Inertia Moment of inertia depends on how far the mass of the object is distributed in relation to the axis of rotation
Table 8-1 on p. 285 gives a list of common moments of inertia
Note that a single object has numerous moments of inertia, since the same object can be rotated in various ways
Examples of Moments of Inertia : Examples of Moments of Inertia
Rotational Equilibrium : Rotational Equilibrium For an object to be in translational equilibrium, its net force must be zero.
For an object to be in rotational equilibrium, its net torque must be zero.
For an object to be in total equilibrium, its net force and net torque must be zero.
Easiest concept, but most difficult problems…
Stop to Think… : Stop to Think… A uniform 6.00 m long horizontal beam that weights 400 N is attached to a wall by a pin connection that allows the beam to rotate. It far end is supported by a cable that makes an angle of 60° with the horizontal, and a 600 N person stands 2.0 m from the pin. Find the tension in the cable, and the reaction force exerted on the beam by the wall, R, if the beam is in equilibrium.
Newton’s 2nd Law for Rotation, Angular Momentum, Rotational Kinetic Energy : Newton’s 2nd Law for Rotation, Angular Momentum, Rotational Kinetic Energy Rotation
Physics
Mr. McCallister
Newton’s 2nd Law for Rotation : Newton’s 2nd Law for Rotation Recall Newton’s Second Law:
F = m a
Force = mass x acceleration
Newton’s Second Law for Rotation states:
t = I a
torque = moment of inertia x angular acceleration
Stop to Think… : Stop to Think… Find the magnitude of a torque produced when a 4.0 N force is applied to a door at a perpendicular distance of 1.0 m away from the hinges.
Angular Momentum : Angular Momentum Recall linear (or translational) momentum:
p = m v
momentum = mass x velocity
Angular momentum is “the product of a rotating object’s moment of inertia and angular speed about the same axis
Symbol: L , Unit: kg m2 / s
L = I ?
angular momentum = moment of inertia x angular speed
Conservation of Angular Momentum : Conservation of Angular Momentum If no net torque is acting on a system, then the system’s angular momentum will be conserved
Example: figure skater’s spin
Decreasing I will yield increased ? to conserve L.
Stop to Think… : Stop to Think… A physics student sitting in an office chair holds two 15 kg dumbbells 1.0 meters away from his center of mass. He starts spinning at 3.0 rad/s, then brings the dumbbells into a distance of 0.3 m away from his center of mass. Find his new angular speed.
Rotational Kinetic Energy : Rotational Kinetic Energy Recall linear (or translational) kinetic energy
KEtrans = ½ m v2
Rotational kinetic energy is due to an object’s rotational motion
KErot = ½ I ?2
rotational kinetic energy = ½ x moment of inertia x (angular speed)2
Conservation of Mechanical Energy : Conservation of Mechanical Energy If only conservative forces (i.e. gravitational, normal) are acting on a system, the system will conserve mechanical energy
However, now ME = KEtrans + KErot + PEg
ME = ½ m v2 + ½ I ?2 + m g h
Stop to Think… : Stop to Think… A bicycle tire of mass 3.0 kg and radius 0.5 m starts rolling from rest on a hill of height 15 m. Find its translational speed at the bottom of the hill.