lecture angular momentum

PHYSICAL SCIENCE PSC1121Physics in Films

© Christos Velissaris & Costas Efthimiou

LECTURE

Angular Momentum

A cannon of mass 100 Kg fires a cannonball of mass 10 Kg with muzzle speed 15m/s. How much is the momentum of the cannon?(All answers are in SI units.)

A 100B 150C 15D 10E There is not enough information to decide

REVIEW QUESTION

In the previous question, what is the recoiling speed of the cannon?(All answers are in SI units.)

A It does not recoil.B 1C 150D 15E 1.5

REVIEW QUESTION

Which of the two children has greater speed ?

A. The girl because she’s at the center

B. The boy because he is at the rim of the wheel and has to travel longer distance

C. Both children have the same speed.

D. It depends on how fast the wheel is spinning

E. Cannot be determined

Assume that in 5 seconds the boy has rotated by a full circle. By what angle has the girl rotated?

In order to describe the rotational motion of a solid, extended object we need something better than the speed. Since different points of the rotating object may have different speed we need something that isthe same for all points of the object. This is the angular velocity, ω.

When an object rotates, we use the angular speed to describe how fast rotates.

timeangle

The angular speed is the ratio of angle (in radians) over the time required to rotate this amount.

Unit of angle:One complete circle = 1 revolution = 360o = 2π rads

rads = radiansrpm = revolutions per minute

Angle θ definition in radians

s (arc length)R

R

Relationship between angular and linear speed in Rotational Motion

A point of a rotating object travels during some time a distance equal to the length of the arc of the circle it follows during its rotation:

or:

Since different points of a rotating object are at different distances from the center of rotation (which is the r) they have different (linear) speed υ. But all points have the same angular speed ω. Points near the center have small r and therefore small υ. Points far from the center of rotation have large r an therefore large υ.

Period and frequency of RotationThe time to takes a rotating object to complete a full revolution(that is to rotate a full circle or 2π radians) is called period and it is denoted by the letter capital T. The angular speed can be written as:

T 2

The inverse of the rotational period is called “frequency” of revolution. It is denoted by the letter f and the number of revolutions (circles) the object rotates per second. The unit of frequency is s-1 (1/s) and it is read inverse seconds. A more popular name is Hertz (or Hz).

Therefore:f

T

Tf

22

1

The SI unit of angular speed is:

A. m/s

B. m/s2

C. rad/s

D. rad/s2

E. none of the above

A CD makes one complete revolutionevery tenth of a second. The angular velocity of point 4 is:A) the same as for pt 1.B) twice that of pt 2.C) half that of pt 2.D) 1/4 that of pt 1.E) four times that of pt 1.

A CD makes one complete revolution every tenth of a second. Which has the largest linear (tangential) velocity?

A) point 1B) point 2C) point 3D) point 4

Which object has the largest magnitude of rotational acceleration?

A. An object starting from rest and reaching 2 rev/s in 20 sec

B. An object rotating at 3 rev/s and coming to a stop in 10 sec

C. An object rotating at a constant 4 rev/s

The equations of the rotational motion remain the same but we need to replace the speed with angular speed and the acceleration with the angular acceleration and the distance with angle.

The equations of the rotational motion remain the same but we need to replace the speed with angular speed and the acceleration with the angular acceleration and the distance with angle.

Newton’s equations for rotational motion remain also the same but we need to also replace the Force with Torque and the Mass with theMoment of Inertia (kind of “rotational mass” ).

Newton’s First law for Rotation: If no torque acts on an object, it either remains still or rotates with constant angular speed.

Newton’s Second law for Rotation: The net torque acting on an object is equal to its Moment of Inertia times its angular acceleration.

TorqueIn order to calculate torque we need two things. First we need a force. Then we need a point to serve as a center of rotation.

The torque of the force with respect to that point (of rotation) is the product of the force with the distance from the point to the line (of action of the force).

Torque of force F with respect to screw location (center of rotation) = force x distance

Which of the forces pictured as acting upon the rod will produce a torque about an axis perpendicular to the plane of the diagram at the left end of the rod?

a) F1b) F2c) Both.d) Neither.

b) F2 will produce a torque about an axis at the left end of the rod. F1 has no lever arm with respect to the given axis.

To increase the applied torque one can

A. Increase the forceB. Increase the lever arm by moving to the end of the leverC. Increase the lever arm by applying the force perpendicularly to the lever D. All of the aboveE. None of the above

Torque of force F with respect to fulcrum (point of rotation)

The two forces in the diagram have the same magnitude. Which orientation will produce the greater torque on the wheel?

a) F1b) F2c) Both.d) Neither.

a) F1 provides the larger torque. F2 has a smaller component perpendicular to the radius.

Torque has a direction (like the force). In order to find its direction we put our finger to the point (of rotation) with respect to which we calculate the torque and we ask ourselves. What is the direction the object would have rotated under the influence of that force? Clockwise or counter-clockwise? This is the direction of the torque.

Usually in physics clockwise rotation is negative (negative torque, negative angular velocity). We say clocks are negative.

Two torques (with respect to the same point of rotation) are opposite if they tend to rotate the object in opposite directions

A 50-N force is applied at the end of a wrench handle that is 24 cm long. The force is applied in a direction perpendicular to the handle as shown. What is the torque applied to the nut by the wrench?

a) 6 N·mb) 12 N·mc) 26 N·md) 120 N·m

b) 0.24 m 50 N = 12 N·m

Newton’s law for rotation application:Where should we put the 3N weight in order to balance the plank?

The 3N weight must generate an equal and opposite torque to the 5N torque with respect to the fulcrum. Only then the net torque is 0?

3 x d = 20 x 5 = 100 or d = 100/3 cm

If two children sit on a seesaw and one is twice as heavy as the other, in order to Achieve balance the lighter child must sit

A) half as far from the fulcrum

B) twice as far from the fulcrum

C) the same distance from the fulcrum

D) four times as far from the fulcrum

How much force F do I need to lift the rock?The torque of my force with respect to fulcrum must be larger than the torque of the rock’s weight. The farther from the fulcrum the less force I need to lift the heavy rock.

When will the plank tip? It will tip when the torque of the man’s weight with respect to the cliff’s edge becomes greater than the torque of the plank’s weight. The farther from the cliff’s edge the man walks the larger its weight’s torque.

When we hang an object with a string the object will stay in such position so that the string line pass from the center of mass of the object. Only then the torque of its weight with respect to the point it is hung from becomes 0

What is the net torque acting on the merry-go-round?

1.2 m 80 N = +96 N·m (counterclockwise)

-(1.2 m 50 N) = -60 N·m (clockwise)

Total (net ) torque = +96 – 60 = + 36 Nm (counterclockwise)

The liquid nitrogen showed in classA is used as fuel in rockets.B is very hot and that’s why we saw vapors

coming out of the container.C is what you get if you freeze the gaseous

nitrogen at very low temperatures.D was really iced water and that’s why the

instructor could touch it with his hand.E is what quality air-conditioning units use to

cool the air.

REVIEW QUESTION

To compare the `reluctance’, or mass, of two objects, accelerate them side-by-side. The one that requires the most force is the

one with the larger mass.

Mass is the measure of the reluctance that an object exhibits in response to any effort made to start it, stop it, or change its state of motion in any way.

Moment of inertia (symbol: I) is the measure of the reluctance that an object exhibits in response to any effort made to change its state of rotational motion.

The moment of inertia depends on two things:

(1) the mass of the object and

(2) the distribution of mass around the axis of rotation.

The formula is I=m d2.

This says that the further the mass from the axis of rotation, the greater the moment of inertia.

A. doubles

B. becomes 4 times the original moment of inertia

C. remains the same

D. halves

E. becomes ¼ of of the original moment of inertia

If we double the distance from the rotation axis, the moment of inertia

I=m d2.

Mass m m

I1 I2

In both pictures the woman has the same massof course. However, the moment of inertia is greater in the first picturesince the hands are further away from the axis of rotation.

Moment of inertia

Two objects, a ring and a disk, both with equal mass and radii. Each is rotating about an axis through its center, as shown. Which has the greatest rotational inertia, I, a uniform disk or a ring?

A) They have the same I, since they both have the same mass and radius

B) The disk

C) The ring

Which rod has the smallest rotational inertia, I, the one rotating about the end, or the one rotating about the center?A) Rotating about end

B) Rotating through center

C) They both have the same I, since they have the same mass and length

Angular Momentumlike momentum, but for rotations

Momentum = Mass x Speed

Angular momentum

Angular speed

Moment of inertia= x

Conservation of Angular Momentumlike conservation of momentum, but for rotations

The law for the conservation of angular momentum states that the angular momentum of a set of objects remains the same if there is no influence on the set of objects from the outside world.

Angular momentum is calculated as the

A) product of mass times velocity

B) product of mass times angular velocity

C)product of moment of inertia times velocity

D)product of moment of inertia times angular velocity

Angular momentum is conserved

A) only when the net force acting on the system is zero

B) only when the angular velocity is zero

C)only when the net torque acting on the system is zero

D)in all cases

The cat uses her tail and the conservation of angular momentum to re-orient her body as she is falling, so she can land on her feet.

A figure skater spins with her arms extended. When she pulls in her arms, she spins faster because

A)she has decreased her rotational inertia.

B) she has increased her rotational inertia.

C) she has increased her angular momentum.

D)the ice exerts a torque on her skates.

Meredith sits in the middle of a merry-go-round that her brother Kiel is pushing. If Meredith moves to the outer edge of the merry-go-round, it will

A) speed up.

B) slow down.

C) spin at the same rate.

If the back propeller was not present, then the helicopter would rotate in the opposite direction with respect the rotor on the top as a result of conservation of momentum. The propeller stops this unwanted rotation.

From Chapter 6: The Rotation of the Rings

play clip

The two rings of the spaceship rotate in opposite directions.

The 6th Day: Arnold pilots a really strange helicopter.

Spy Kids 2: The helicopter is strange but the physics is good.

Have you understood all this?

Imagine that a highway is built to join US to the NORTH POLE. Also, imagine now that all people in North and South America decide to drive their cars to the North Pole. Then, what of the following would happen?

A. The rotation of Earth will slow down.

B. The rotation of Earth will speed up.

C. Nothing will happen.

D. The Earth will flip up-side down.

Torque: Motorcycle games.

Under what conditions can this happen?

Speed: A nasty turn.