Chapter_11

# Chapter_11 - Chapter 11 Rolling Torque and Angular Momentum...

This preview shows pages 1–6. Sign up to view the full content.

Chapter 11 Rolling, Torque, and Angular Momentum In this chapter we will cover the following topics: - Rolling of circular objects -Redefinition of torque to describe rotational problems that are more complicated than the rotation of a rigid body about a fixed axis - Angular momentum of single particle and a system of particles - Conservation of angular momentum --Applications of the conservation of angular momentum (11-1)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
t 1 = 0 t 2 = T Consider a circular object that rolls along a surface without slipping. We can simplify its study by treating it as a combination of translation of COM and Rolling as Translation and Rotation Combined rotation about COM Consider two snapshots of a rolling bicycle wheel. An observer on the ground sees the COM (point O) of the wheel move forward with a speed . During the time interval between the two snapshots p com v T oint O covers a distance (eqs.1) During time interval T, the bicycle rider sees the wheel rotate by an angle about O, so (eqs.2). If we equation 1 with equation 2, we g = = = com s v T s R TR combine θ θ ϖ et the condition for rolling without slipping. com v R ϖ = (11-2)
Rolling is a combination of a pure translation with speed and a pure rotation about COM with angular velocity . The velocity of each point is the vector sum of the velocities of the two m = com com v v R ϖ otions. For the translational motion, the velocity is the same for every point ( ). The rotational velocity varies from point to point. Its magnitude is equal to where is the distance from com v r r ϖ r P T O. Its direction is tangent to the circular orbit. The net velocity is the vector sum of these two terms. For example, v = 0 v = 0 = 2 . - = + = + = com com O com com com com com v v v v v v v v r r r r r r r r r r com v R ϖ = (11-3)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
A B v A v B v T v O We consider rolling as a pure rotation about an axis that passes through the contact point P between the wheel and the road. The angular velocity of the rot Alternative view : Rolling as Pure Rotation ation is The rotation axis at P changes with time = Note : com v R ϖ Let's find the velocity vector v for each point on the wheel. The direction of v points along the tangent to the circular orbit. For example, at point A, is perpendicular to the line AP. The A v r r r magnitude of v is , where is the distance between a given point and the contact point P. For example, At point T 2 2 2 . At point O . At point P 0 = = = = = = = = T com O com P v r r r R v R v r R v R v r v ϖ ϖ ϖ r 0 = (11-4)
For a rolling object (mass and radius ) we calculate its kinetic energy by considering rolling as a pure rotation about the contact point P. The Kinetic Energy of Rolling M R ( 29 2 2 2 2 2 2 1 The kinetic energy is . Here is the rotational inertia of the rolling 2 body about point P.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern