5/6/2018. Rolling Without Slipping. Rolling Without Slipping. QuickCheck 12.10

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Rolling Without Slipping Rolling is a combination of rotation and translation.

the angular velocity by Slide 12-93 Rolling Without Slipping The figure below shows how the

v bottom = 0. The point on the bottom of a rolling object is instantaneously at rest.

1 Rolling Without Slipping Rolling is a combination of rotation and translation. For an object that rolls without slipping, the translation of the center of mass is related to the angular velocity by Slide Rolling Without Slipping The figure below shows how the velocity vectors at the top, center, and bottom of a rolling wheel are found. v top = 2v cm. v bottom = 0. The point on the bottom of a rolling object is instantaneously at rest. Slide QuickCheck A wheel rolls without slipping. Which is the correct velocity vector for point P on the wheel? Slide

Kinetic Energy of Rolling The kinetic energy of a rolling object is In other words, the rolling motion of a rigid body can be described as a translation of the center of mass (with kinetic energy K

2 Kinetic Energy of Rolling The kinetic energy of a rolling object is In other words, the rolling motion of a rigid body can be described as a translation of the center of mass (with kinetic energy K cm ) plus a rotation about the center of mass (with kinetic energy K rot ). Slide Figure Example 1 - Which object makes it down the ramp the fastest? Figure Example 2 - An 8.0-cm-diameter, 400g solid sphere is released from rest at the top of a 2.1-m-long, 25 degree incline. It rolls, without slipping, to the bottom. a) What is the sphere s angular velocity at the bottom of the incline? b) What fraction of its kinetic energy is rotational? 2

The Vector Description of Rotational Motion One-dimensional motion uses a scalar velocity v and force F. A more general understanding of motion requires vectors and.

Slide 12-99 The Angular Velocity Vector The magnitude of the angular velocity vector is.

3 The Vector Description of Rotational Motion One-dimensional motion uses a scalar velocity v and force F. A more general understanding of motion requires vectors and. Similarly, a more general description of rotational motion requires us to replace the scalars and with the vector quantities and. Doing so will lead us to the concept of angular momentum. Slide The Angular Velocity Vector The magnitude of the angular velocity vector is. The angular velocity vector points along the axis of rotation in the direction given by the right-hand rule as illustrated. Slide The Cross Product of Two Vectors The dot product is one way to multiply two vectors, giving a scalar. A different way to multiple two vectors, giving a vector, is called the cross product. If vectors and have angle between them, their cross product is the vector Slide

The Right-Hand Rule The cross product is

The right-hand rule for the direction comes in

Try them all to see which works best for you.

18 Calculating a Cross Product Slide 12-103

4 The Right-Hand Rule The cross product is perpendicular to the plane of and. The right-hand rule for the direction comes in several forms. Try them all to see which works best for you. Note that. Instead,. Slide EXAMPLE Calculating a Cross Product Slide The Torque Vector We earlier defined torque τ = rfsinϕ. r and F are the magnitudes of vectors, so this is a really a cross product: A tire wrench exerts a torque on the lug nuts. Slide

Figure 12.45 Example 3 A force ԦF = 11jƸ N is exerted on a particle at Ԧr = 8i Ƹ + 5jƸ m. What is the torque on the particle about the origin? Find ԦA B given ԦA = 2i Ƹ + 3jƸ and B = 3iƸ 2j Ƹ + 2k.

5 Figure Example 3 A force ԦF = 11jƸ N is exerted on a particle at Ԧr = 8i Ƹ + 5jƸ m. What is the torque on the particle about the origin? Find ԦA B given ԦA = 2i Ƹ + 3jƸ and B = 3iƸ 2j Ƹ + 2k. Angular Momentum of a Particle A particle of mass m is moving. The particle s momentum vector makes an angle with the position vector. We define the particle s angular momentum vector relative to the origin to be Slide Angular Momentum of a Particle Why this definition? If you take the time derivative of and use the definition of the torque vector (see book for details), you find: Torque causes a particle s angular momentum to change. This is the rotational equivalent of and is a general statement of Newton s second law for rotation. Slide

Figure 12.45 Example 4 A satellite follows the elliptical orbit as shown. The only force on the satellite is the gravitational attraction of the planet. The satellite's speed at point a is 7200 m/s.

6 Figure Example 4 A satellite follows the elliptical orbit as shown. The only force on the satellite is the gravitational attraction of the planet. The satellite's speed at point a is 7200 m/s. a) What is the net torque at point a? b) What is the satellite s speed at point b and c? Angular Momentum of a Rigid Body For a rigid body, we can add the angular momenta of all the particles forming the object. If the object rotates on a fixed axle, or about an axis of symmetry then it can be shown that And it s still the case that. Slide Analogies Between Linear and Angular Momentum and Energy Slide

7 Conservation of Angular Momentum An isolated system that experiences no net torque has and thus the angular momentum vector constant. is a Slide Conservation of Angular Momentum As an ice skater spins, external torque is small, so her angular momentum is almost constant. By drawing in her arms, the skater reduces her moment of inertia I. To conserve angular momentum, her angular speed must increase. Slide QuickCheck Two buckets spin around in a horizontal circle on frictionless bearings. Suddenly, it starts to rain. As a result, A. The buckets speed up because the potential energy of the rain is transformed into kinetic energy. B. The buckets continue to rotate at constant angular velocity because the rain is falling vertically while the buckets move in a horizontal plane. C. The buckets slow down because the angular momentum of the bucket + rain system is conserved. D. The buckets continue to rotate at constant angular velocity because the total mechanical energy of the bucket + rain system is conserved. E. None of the above. Slide

8 EXAMPLE Two Interacting Disks Slide EXAMPLE Two Interacting Disks Slide

PHYSICS. Chapter 12 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc.

PHYSICS. Chapter 12 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc. PHYSICS FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E Chapter 12 Lecture RANDALL D. KNIGHT Chapter 12 Rotation of a Rigid Body IN THIS CHAPTER, you will learn to understand and apply the physics