Chapter 8 – Rotational equilibrium and dynamics

 

8-1 Torque

 

torque – a quantity that measures the ability of a force to rotate an object around some axis

 

·      rotational motion and translational motion can be treated separately

 

·      torque depends on

o    amount of force

o    distance from pivot

o    angle of applied force

 

lever arm – the perpendicular distance from the axis of rotation to a line drawn along the direction of the force

 

Τ = Fd sinθ

Τ – the Greek letter “tau”

d – the length of the lever arm

F sinθ – the amount of force in the direction                    perpendicular to the lever arm

Torque is a vector quantity

·      magnitude and direction

·      found by the cross product of 2 vectors

            F and d

·      Τ = ||F|| ||d|| sinθ

            where θ is the angle between the                         vectors F and d

 

·      Cross products produce a vector perpendicular to both original vectors.

·      The sign of the new vector is determined by the “right hand rule”.

 

For our purposes…

ü …if the object rotates counterclockwise then give it a POSITIVE sign

ü …if the object rotates clockwise then give it a NEGATIVE sign

 

If more than one force is present, each force will produce a torque.

      ΣΤ = Τnet

…so calculate each torque – watch the sign of each – and then add.

 

Example:  A student pushes with a minimum force of 50.0 N on the middle of a door to open it.

a.  What minimum force must be applied at the far edge of the door in order for the door to open?

 

 

b.  What minimum force must be applied to the hinged side of the door in order for the door to open?

 

Example:  A bucket filled with water has a mass of 54 kg and is hanging from a rope that is wound around a 0.050 m radius stationary cylinder.  If the cylinder does not rotate and the bucket hangs straight down, what is the magnitude of the torque the bucket produces around the center of the cylinder?

8-2 Rotation and inertia

 

center of mass – the point at which all the mass of a body can be considered to be concentrated when analyzing translational motion

 

center of gravity – the point at which the gravitational force acts on the extended object as if it were a point mass

 

*in a uniform gravitational field, the center of mass and the center of gravity are equivalent

 

moment of inertia – the tendency of a body rotation about a fixed axis to resist a change in rotational motion

·      analogous to mass

·      depends on mass AND the distribution of the mass around the axis of rotation

·      the farther the mass (on average) is from the axis of rotation, the greater the moment of inertia – and it is more difficult to rotate the object

 

mass far from axis               mass close to axis

hard to turn                           easy to turn

large inertia                          small inertia

 

So…consider a race with a solid sphere and a ring (hoop) with equal radii and mass.  Which would roll down an incline faster?

 

p. 285 – chart of moments of inertia

         - use the letter I   

         - unit:  kgm2

         - formulas come from integrals

 

rotational equilibrium – the net torque on an object is zero

 

translational equilibrium – the net force on an object is zero

 

CONDITIONS FOR EQUILIBRIUM

1.  ΣF = 0     net force is zero

2.  ΣΤ = 0     net torque is zero

 

*If ΣF = 0, can the object be moving?

Yes! Moving at a constant speed.

 

*  If ΣΤ = 0, can the object be rotating?

Yes! Rotating at a constant speed.

How to find ΣΤ

1.  choose an axis of rotation

         - anywhere – but preferably where one                or more torques is zero

2.  calculate each torque and assign a sign (+,-) depending on which way the object would rotate if that one force were the only one applied to the object

3.  solve

 

Example:  A uniform 5.00 m long horizontal beam that weighs 315 N is attached to a wall by a pin connection that allows the beam to rotate.  Its far end is supported by a cable that makes an angle of 53° with the horizontal, and a 545 N person is standing 1.50 m from the pin.  Find the force in the cable, FT, and the force exerted on the beam by the wall, R, if the beam is in equilibrium.

 

8-3 Rotational dynamics

 

Newton’s 2nd Law for rotating objects

 

   Τnet = I α

net torque = moment of inertia x angular acc.

 

unit:  Nm

 

Example:  A disk with mass of 165.0 g and a radius of 13.5 cm that is spinning at 30 rad/s can be stopped in 0.10s.  What is the average torque on the disk by the hand?