Chapter 11. Today. Last Wednesday. Precession from Pre- lecture. Solving problems with torque

Size: px

Start display at page:

Download "Chapter 11. Today. Last Wednesday. Precession from Pre- lecture. Solving problems with torque"

Francine Ball
6 years ago
Views:

1 Chapter 11 Last Wednesday Solving problems with torque Work and power with torque Angular momentum Conserva5on of angular momentum Today Precession from Pre- lecture Study the condi5ons for equilibrium of a body using FBDs, forces and torques. Discuss center of gravity and how it relates to a body s stability Solve problems for rigid bodies in equilibrium (We are only covering 11.1 è 11.3)

2 Clicker Ques5on Correc5on A student holding a ball sits on the outer edge a merry go round which is ini5ally rota5ng counterclockwise. Which way should she throw the ball so that she stops the rota5on? A) To her les B) To her right C C) Radially outward ω A B top view: ini5al final

Prelecture L vector and precession; ques5on 1 A student is ini5ally at

plane. The student is holding a wheel spinning as shown in (1).

(2). If he keeps turning the wheel over in the same direc5on un5l it ends

3 Prelecture L vector and precession; ques5on 1 A student is ini5ally at rest on a stool that can rotate freely without fric5on in the horizontal plane. The student is holding a wheel spinning as shown in (1). He turns the wheel over and as a result he and the stool start to rotate (2). If he keeps turning the wheel over in the same direc5on un5l it ends up in its original orienta5on (3), his rota5on will... a) stop. b) double. c) stay the same.

4 Prelecture L vector and precession; ques5on 2 A disk is spinning with angular velocity ω on a pivoted horizontal axle as shown. Gravity acts down and the disk has a precession frequency Ω. If the angular velocity of the disk were doubled, the precession frequency would... a) increase. b) decrease. c) stay the same.

5 Prelecture L vector and precession; Checkpoint 1 A disk is spinning with angular velocity ω on a pivoted horizontal axle as shown. Gravity acts down. In which direc5on does precession cause the disk to move? a) Into the page b) Out of the page c) Up d) Down

6 Prelecture L vector and precession; Checkpoint 2 A disk is spinning with angular velocity ω on a pivoted horizontal axle as shown. Gravity acts down and the disk has a precession frequency Ω. If the mass of the disk were doubled but its radius and angular velocity were kept the same, the precession frequency would a) increase. b) decrease. c) stay the same.

7 Prelecture L vector and precession; Checkpoint 3 If the radius of the disk were doubled but its mass and angular velocity were kept the same, the precession frequency would a) increase. b) decrease. c) stay the same.

8 Chapter 11

Equilibrium for rigid bodies For a finite- sized object to be in equilibrium, all the forces and torques must cancel!!f = 0 and!!! = 0 which is equivalent to these 3 equa5ons! F x = 0! F y = 0!

9 Equilibrium for rigid bodies For a finite- sized object to be in equilibrium, all the forces and torques must cancel!!f = 0 and!!! = 0 which is equivalent to these 3 equa5ons! F x = 0! F y = 0!! = 0 Since the body is in equilibrium (not moving) there isn t really an axis of rota5on. So you can choose whichever axis is the most convenient, or simplifies the equa5ons the most. The center of gravity is equal to the center of mass, as long as the value of g doesn t change much over the size of the object.

10 Torque due to gravity R Magnitude: τ = R CM Mg sin (θ ) = MgR R CM Lever arm

11 Prelecture ques5on 1 In all three cases shown below the beam has the same mass and length and is acached to the wall by a hinge. Gravity acts downward. In which case is the torque due to gravity on the beam about an axis through the hinge the biggest?

12 Prelecture ques5on 2 In both cases shown below the horizontal beam has the same mass and length and is acached to the wall by a hinge. In each case a wire runs from the end of the beam to the wall. In which case is the tension in the wire greatest?

In case 2 the massless rod is twice as long and makes an angle of 30o with the wall as

13 Checkpoint ques5on 1 In case 1, one end of a horizontal massless rod of length L is acached to a ver5cal wall by a hinge, and the other end holds a ball of mass M. In case 2 the massless rod is twice as long and makes an angle of 30o with the wall as shown. In which case is the total torque about an axis through the hinge biggest? a) Case 1 b) Case 2 c) Same

The object is then pivoted at a point a distance L/4 from the end of the plank

14 Checkpoint ques5on 2 An object is made by hanging a ball of mass M from one end of a plank having the same mass and length L. The object is then pivoted at a point a distance L/4 from the end of the plank suppor5ng the ball, as shown below. Is the object balanced? a) Yes b) No, it will fall to the les. c) No, it will fall to the right.

Checkpoint ques5on 3 In case 1, one end of a horizontal plank of mass M and length L is acached to a wall by a hinge and the other end is held up by a wire acached to the wall.

15 Checkpoint ques5on 3 In case 1, one end of a horizontal plank of mass M and length L is acached to a wall by a hinge and the other end is held up by a wire acached to the wall. In case 2 the plank is half the length but has the same mass as in case 1, and the wire makes the same angle with the plank. In which case is the tension in the wire biggest? a) Case 1 b) Case 2 c) Same

16 Clicker Ques5on Which equa5on correctly expresses that the total torque about the hinge is zero? a) TL! Mg L 2 = 0! b) TLsin!! Mg L 2 = 0 T c) T L 2! Mgsin! = 0 θ d) TL! MgLsin! = 0 Mg L

17 Problem solving approach 1. Draw a picture 2. Pick a body that is in equilibrium and draw all forces ac5ng on it, include the posi5on of each force vector 3. Choose x- y- coordinates and an axis of rota5on. Define the posi5ve sense of rota5on (C- C- W around axis)! F y = 0 4. Set up F x = 0,, and 5. Solve for unknowns!!! = 0

18 Example Consider a plank of mass M suspended by two strings as shown. What is the tension in each string? T 1 T 2 x cm L/2 L/4 M Mg y x

19 Example cont. Consider a plank of mass M suspended by two strings as shown. What is the tension in each string? T 1 T 2 x cm L/2 L/4 M Mg y x

20 Example Hanging beam A purple beam is hinged to a wall holding up a blue sign. The beam has a mass of m b = 6.00kg and the sign of mas m s = 17.0kg. The length of the beam is L = 2.81m. The sign is acached at the end of the beam and the horizontal wire holding is acached 2/3 of the way to the end of the beam. The angle the wire makes with the beam is θ = a) What is the tension in the wire? b) What is the net force the hinge exerts on the beam? c) The maximum tension the wire can have without breaking is T = 891N. What is the maximum mass sign that can be hung from the beam? y x m s g T m b g H y H x

21 Clicker Ques5on A metal adver5sing sign (weight w) is suspended from the end of a massless rod of length L. The rod is supported at one end by a hinge at point P and at the other end by a cable at an angle θ from the horizontal. What is the tension in the cable? A. T = w sin θ B. T = w cos θ C. T = w/(sin θ) D. T = w/(cos θ) E. none of the above

A fric5onal force f between the ladder and the floor keeps it from slipping.

22 Example balanced ladder Bill (mass m) is climbing ladder (length D and mass M l ) that leans against a smooth wall. A fric5onal force f between the ladder and the floor keeps it from slipping. The angle between the ladder and the floor is θ. What is the dependence of f on the angle of the ladder and Bill s distance up the ladder? D φ Bill m M l y x f θ

Example Using r! instead of F! You are trying to raise a bicycle wheel of mass m and radius R up over a curb of height h. To do this, you apply a horizontal force F.

23 Example Using r! instead of F! You are trying to raise a bicycle wheel of mass m and radius R up over a curb of height h. To do this, you apply a horizontal force F. What is the smallest magnitude of the force that will succeed in raising the wheel onto the curb when the force is applied a) at the center of the wheel? b) at the top of the wheel? c) which is greater?

24 Where is the normal force? The normal force and the fric5onal force will be directly under the center of gravity of the object. When the center of gravity is outside of the footprint of the object it is no longer stable, and will 5p, lowering its poten5al energy. θ θ

25 Clicker Ques5on Suppose you hang one end of a beam from the ceiling by a rope and the bocom of the beam rests on a fric5onless sheet of ice. The center of mass of the beam is marked with an black spot. Which of the following configura5ons best represents the equilibrium condi5on of this setup? A) B) C)

Last Time: Finish Ch 9 Start Ch 10 Today: Chapter 10

Last Time: Finish Ch 9 Start Ch 10 Today: Chapter 10 Monday Ch 9 examples Rota:on of a rigid body Torque and angular accelera:on Today Solving problems with torque Work and power with torque Angular momentum