10/24/2010
1
Gravitational force: example
A satelite is placed between two planets at rest. The mass of the red planet is 4 times that of the green. The distance of the satelite to the green is twice that of the red. What happens to the satelite?
a) Moves toward the greenb) Moves toward the redc) Stays at rest
Rotational equilibrium and dynamics
10/24/2010
2
Torque
The tendency of a force to rotate an object about some axis is measured by a quantity called torque Fd=τUnits Nm1 rF φτ sin=
Demo openning a door: question time
A door has 3 knobs. Which one would require a larger force to open the door?
A B C
hinge
10/24/2010
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Torque: question time
A force is applied on a rotating disk. When will the torque be maximum?
a) When the force is applied along the tangentb) When the force is pointing toward the center (radially)c) When the force is applied at an angle of 45 degreesd) The torque is always the same
A force is applied on a rotating disk. When will the torque be zero?
a) When the force is applied along the tangentb) When the force is pointing toward the center (radially)c) When the force is applied at an angle of 45 degreesd) The torque is always the same
Torque: example
Find the torque produced by the 300 N force applied to a door at an angle of 60 deg.
10/24/2010
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Centre of gravity
Consider a system of N little particles. The center of gravity of this system is given by:
N
NNcm mmm
xmxmxmx
++++++=
...
...
21
2211
N
NNcm mmm
ymymymy
++++++=
...
...
21
2211
Centre of gravity: question time
Which of the following statements is correct:a) The center of gravity is the
geometric center therefore the center of mass is x=0 y=0
b) The xCM > 0 because there is more mass on the right than on the left
c) The center of mass is only determined by the heavy masses
d) There is not enough information to determine the center of mass of the system.
10/24/2010
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Centre of gravity
Consider a system of 4 balls The center of gravity is:
4232
24)2(2)2(322
+++×+−×+−×+×=cmx
4232
34323332
+++×−×−×+×=cmy
Centre of gravity: example
Three particles are located in a coordinate system as shown in the figure. Find the centre of gravity.
10/24/2010
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Two conditions for equilibrium
Equilibrium under translation: an object is in equilibrium if the net force applied to it is zero
0=Σ= inet FFrr
Equilibrium under rotation: an object is in equilibrium if
the net torque applied to it is zero
0=Σ= inet ττ rr
Strategy for Objects in Equilibrium
� draw diagram for the system� isolate the object � draw the free-body diagram showing all external forces (if there is more than one body, draw a diagram for each object)� define a coordinate system to decompose forces� apply the first condition of equilibrium along x and y� define a rotation axis for calculating the torque� apply the second condition of equilibrium� solve the set of simultaneous equations to determine the unknowns
10/24/2010
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Two conditions for Equilibrium: example
An arm holds a weight according to the figure. Determine the upward force exerted by the biceps on the forearm (F) and the downward force exerted on the joint ( R).
Two conditions for equilibrium: example
A uniform horizontal beam, 5.00 m long, weighing 3.00 x 102 N, is attached to a wall by a pin connection, that allows the beam to rotate. Its far end is supported by a cable as shown in the figure. A person, weighing 6.00 x 102 N, stands 1.50 m away from the wall on the beam. Find the magnitude of the tension on the cable and the reaction force of the wall on the beam.
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Two conditions for equilibrium: example
A uniform ladder 10.0 m long and weighing 50.0 N rest against a smooth vertical wall. If the ladder is just on the verge of slipping when it makes 50.0 degree angle with the ground, find the coefficient of friction between the ladder and the ground.
0.8 m
Torque and angular acceleration
The tangential force acting on an object is related to the angular acceleration in the following way:
)( αrmmaFt ==
ατ )( 2mrFr t ==The torque is then:
Moment of inertia
10/24/2010
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Torque of a rotating object
ααττ Imriinet === ∑∑ )( 2
The net torque applied to an object is:
∑= 2imrI
The moment of inertia is an intrinsic property of the systemrelated with the mass distribution around the axis of rotation
Units 21 mkg
Moment of inertia: question time
A constant net torque is applied to an object. Which of these will definitely not be constant:
a) angular acceleration, b) angular velocity, c) moment of inertia, d) center of gravity.
Two cylinders have the same mass and radius, and are rolling down an incline. One is hollow, the other is not. Which one arrives at the bottom first?
10/24/2010
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Moment of inertia
Moment of inertia: question time
Two cylinders have the same mass and radius, and are rotating with the same constant angular velocity. One is hollow, the other is not. If the same breaking torque is applied, which one takes longer to stop?
Two bar demo
10/24/2010
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Moment of inertia: question time REPEAT
Two cylinders have the same mass and radius, and are rotating around the axis of symmetry with the same constant angular velocity. One is hollow, the other is not. If the same breaking torque is applied, which one takes longer to stop?
Two cylinders have the same mass and radius, and are rolling down an incline. One is hollow, the other is not. Which one arrives at the bottom first?
Moment of inertia: example
A majorette twirls an unusual baton made of four spheres. Each rod is 1.0 m long. Find the moment of inertia of the system:
a) For rotations about the axis of symmetry perpendicular to the page and passing through the points where the rods cross?
b) For rotations about the axis OO’
10/24/2010
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Two conditions for dynamics of a rigid body
Translation: the acceleration of an object is proportional to the net force applied
amFF inet
rrr=Σ=
Rotation: the angular acceleration of an object is proportional to the net torque applied
αττ vrrIinet =Σ=
Dynamics of a rigid body: example
A frictionless solid reel of M=3.00 Kg and R=0.400 m is used to draw water from a well. A bucket m=2.00 kg is attached to the cord, wrapped around the cylinder.
a) Find the Tension on the cord and the acceleration of the bucket.
b) If the bucket starts at rest from the top of the well and falls for 3.00 s before hitting the water, how far does it fall?
10/24/2010
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Rotational kinetic energy
Even if the center of gravity of the object is at rest, a rotating object has kinetic energy!
2
2
1 ωIKEr =
The total kinetic energy of the object is the sum of the translational and the rotational kinetic energies
22
2
1
2
1 ωImvKEKEKE rttot +=+=
Rotational kinetic energy: question time
Two spheres, with the same mass and radius, are rotating with the same angular speed about the symmetry axis. The first is hollow and the second is not. How do their kinetic energies relate?
a) KE1=KE2 b) KE1<KE2 c) KE1>KE2
A ball and a box have the same mass. Which one will arrive first at the bottom of an incline
a) the ball rolling without sliding down the incline?b) the box sliding down the incline?
A ball and a cylinder have the same mass. Which one will arrive first at the bottom of an incline
a) a ball rolling without sliding down the incline?b) a cylinder rolling without sliding down the incline?
10/24/2010
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Rotational energy: example
A ball of mass M and radius R, starts from rest at the top of an incline and rolls down. What is its linear speed at the bottom of the incline?
The angular momentum of a particle in circular motion is proportional to the mass, the radius of the orbit and its speed:
Angular momentum in circular motion
mvRL =
vR
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The angular momentum of a particle relative to a rotation axis is proportional to the mass, the shortest distance from the axis of rotation of the path of the straightline orbit and its speed:
Angular momentum in collisions
mvbL =
vb
Angular momentum of a rigid object
The angular momentum of a rigid object is proportional to its moment of inertia and the angular speed
ωIL =The torque acting on an object is equal to the time rate of change of the object’s angular momentum
t
L
∆∆=τ
ταωωωωω ≡=∆∆=
∆−=
∆−=
∆∆
It
It
It
II
t
L 00
Derivation:
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Conservation of angular momentum
The angular momentum of a system is conserved when the net external torque acting on the system is zero.
fi LL =
ffii II ωω =
Conservation of angular momentum: question
A horizontal disk with moment of inertia I1 rotates with an initial angular speed about a vertical axis. A second horizontal disk, with moment of inertia I2, initially not rotating, drops onto the first and sticks to it. What is the ratio between the final and initial angular speeds?
a) I1/I2b) I2/I1c) I1/(I1+I2)d) I2/(I1+I2)
10/24/2010
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Conservation of angular momentum: question
A student is playing with weights on a rotating stool. Assume there is no friction. If initially he has his arms stretched, what happens when he brings them in?
a) The angular speed decreasesb) The angular speed increasesc) The angular speed stays the same
translation versus rotation
2
2
1 ωIKEr =2
2
1MvKEt =
t
pF
∆∆=
t
L
∆∆=τ
maF = ατ I=
ωIL =Mvp =
10/24/2010
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Conservation of Angular momentum: example
You jump on a merry go round which is a simple circular platform of mass M=100 kg and radius R=2.00 m. When you are standing right at the edge the angular speed of the system is 2.00 rad/s. You slowly walk toward the centre. What is the angular speed of the system when you reach a point 0.500 m from the centre?
