Gravitational force: example

Transcription

1 10/4/010 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 green b) Moves toward the red c) Stays at rest Rotational equilibrium and dynamics 1

2 10/4/010 Torque The tendency of a force to rotate an object about some axis is measured by a quantity called torque Units 1 Nm τ = τ = Fd F sinφ r 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

3 10/4/010 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 tangent b) When the force is pointing toward the center (radially) c) When the force is applied at an angle of 45 degrees d) 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 tangent b) When the force is pointing toward the center (radially) c) When the force is applied at an angle of 45 degrees d) 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. 3

4 10/4/010 Centre of gravity Consider a system of N little particles. The center of gravity of this system is given by: x y cm cm m1x1 mx... m = m m... m 1 m1 y1 m y... m = m m... m 1 N N N N x y N N 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 x CM > 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. 4

5 10/4/010 5 Centre of gravity Consider a system of 4 balls The center of gravity is: ) ( ) ( 3 = x cm = y cm Centre of gravity: example Three particles are located in a coordinate system as shown in the figure. Find the centre of gravity.

6 10/4/010 Two conditions for equilibrium Equilibrium under translation: an object is in r r F net = ΣFi equilibrium if the net force applied to it is zero = 0 Equilibrium under rotation: an object is in equilibrium if the net torque applied to it is zero r r τ net = Στ i = 0 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 6

7 10/4/010 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 10 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 10 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. 7

8 10/4/010 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: F t = ma = m( rα) The torque is then: τ = r Ft = ( mr )α Moment of inertia 8

9 10/4/010 Torque of a rotating object The net torque applied to an object is: τ net = = τ i = ( mr i ) α Iα The moment of inertia is an intrinsic property of the system related with the mass distribution around the axis of rotation Units 1 kg m I = mr i 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? 9

10 10/4/010 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

11 10/4/010 Moment of inertia: question time REPEAT 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? 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? 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 11

12 10/4/010 Two conditions for dynamics of a rigid body Translation: the acceleration of an object is proportional to the net force applied r F net r = ΣF i r = ma Rotation: the angular acceleration of an object is proportional to the net torque applied r τ r = Στ net i = v Iα 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=.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? 1

13 10/4/010 Rotational kinetic energy Even if the center of gravity of the object is at rest, a rotating object has kinetic energy! 1 KEr = Iω The total kinetic energy of the object is the sum of the translational and the rotational kinetic energies KE 1 tot = KEt KEr = mv 1 Iω 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) KE 1 =KE b) KE 1 <KE c) KE 1 >KE 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? 13

14 10/4/010 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? Angular momentum in circular motion The angular momentum of a particle in circular motion is proportional to the mass, the radius of the orbit and its speed: L = mvr v R 14

15 10/4/010 Angular momentum in collisions 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: v b L = mvb Angular momentum of a rigid object The angular momentum of a rigid object is proportional to its moment of inertia and the angular speed L = Iω The torque acting on an object is equal to the time rate of change of the object s angular momentum Derivation: τ = L t t Iω Iω0 ω ω = = I t t L 0 = I ω = t Iα τ 15

16 10/4/010 Conservation of angular momentum The angular momentum of a system is conserved when the net external torque acting on the system is zero. L = i L f I ω = i i I f ω f Conservation of angular momentum: question A horizontal disk with moment of inertia I 1 rotates with an initial angular speed about a vertical axis. A second horizontal disk, with moment of inertia I, initially not rotating, drops onto the first and sticks to it. What is the ratio between the final and initial angular speeds? a) I 1 /I b) I /I 1 c) I 1 /(I 1 I ) d) I /(I 1 I ) 16

17 10/4/010 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 decreases b) The angular speed increases c) The angular speed stays the same p = Mv translation versus rotation L = Iω F = F = ma p t τ = τ = Iα L t KEt = 1 Mv KEr = 1 Iω 17

18 10/4/010 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=.00 m. When you are standing right at the edge the angular speed of the system is.00 rad/s. You slowly walk toward the centre. What is the angular speed of the system when you reach a point m from the centre? 18