Today: (Ch. 8)  Rotational Motion

Center of Gravity For the purposes of calculating the torque due to the gravitational force, you can assume all the force acts at a single location The location is called the center of gravity of the object The center of gravity and the center of mass of an object are usually the same point

Rotational Equilibrium Equilibrium may include rotational equilibrium An object can be in equilibrium with regard to both its translation and its rotational motion Its linear acceleration must be zero and its angular acceleration must be zero The total force being zero is not sufficient to ensure both accelerations are zero

Equilibrium Example The applied forces are equal in magnitude, but opposite in direction Therefore, ΣF = 0 However, the object is not in equilibrium The forces produce a net torque on the object There will be an angular acceleration in the clockwise direction For an object to be in complete equilibrium, the angular acceleration is required to be zero Στ = 0 This is a necessary condition for rotational equilibrium All the torques will be considered to refer to a single axis of rotation The same ideas can also be applied to multiple axes

Rotational Equilibrium, Lever Use rotational equilibrium to find the force needed to just lift the rock We can assume that the acceleration is zero Also ignore the mass of the lever The force exerted by the person can be less than the weight of the rock The lever will amplify the force exerted by the person If Lperson > Lrock

Tipping a Crate We can calculate the force that will just cause the crate to tip When on the verge of tipping, static equilibrium applies If the person can exert about half the weight of the crate, it will tip

Moment of Inertia The moment of inertia composed of many pieces of mass is The moment of inertia of an object depends on its mass and on how this mass is distributed with respect to the rotation axis The definition can be applied to find the moment of inertia of various objects for any rotational axis Units of moment of inertia are kg · m2

Various Moments of Inertia

Rotational Dynamics Newton’s Second Law for a rotating system states Once the total torque and moment of inertia are found, the angular acceleration can be calculated Then rotational motion equations can be applied For constant angular acceleration:

Kinematic Relationships

Real Pulley with Mass, Example Up to now, we have assumed a massless pulley Using rotational dynamics, we can deal with real pulleys The torque on the pulley is due to the tension in the rope Apply Newton’s Second Laws for translational motion and for rotational motion The crate undergoes translational motion The pulley undergoes rotational motion For the pulley: The tension in the rope supplies the torque The pulley rotates around its center, so that is a logical axis of rotation

Motion of a Crate, Example, cont Pulley equation Στ = - T R = Ipulley α The pulley is a disc, so I = ½ mpulley R²pulley For the crate Take the +y direction as + Equation: ΣF = T – mcrate g = mcrate a Relating the accelerations a = α Rpulley Combine the equations and solve

Example If the mass of a wheel is increased by a factor of 17 and the radius is increased by a factor of 9, by what factor is the moment of inertia increased? A factor of

Example If the mass and height of the object is increased by twice, what would be the increase/decrease of final potential energy?

Example If final velocity is increased by 3 times, what would be increase/decrease in final kinetic energy?

Tomorrow: (Ch. 9)  Energy and Momentum of Rotational Motion