Physics
Rolling Motion
Rolling motion refers to the movement of an object as it rotates and translates simultaneously. This type of motion combines both rotational and translational kinetic energy. When an object rolls, it experiences both linear and angular velocity, allowing it to move forward while also rotating about its axis.
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10 Key excerpts on "Rolling Motion"
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
310 C H A P T E R 1 1 Rolling, Torque, and Angular Momentum What Is Physics? As we discussed in Chapter 10, physics includes the study of rotation. Arguably, the most important application of that physics is in the Rolling Motion of wheels and wheel-like objects. This applied physics has long been used. For example, when the prehistoric people of Easter Island moved their gigantic stone statues from the quarry and across the island, they dragged them over logs acting as rollers. Much later, when settlers moved westward across America in the 1800s, they rolled their possessions first by wagon and then later by train. Today, like it or not, the world is filled with cars, trucks, motorcycles, bicycles, and other rolling vehicles. The physics and engineering of rolling have been around for so long that you might think no fresh ideas remain to be developed. However, skateboards and inline skates were invented and engineered fairly recently, to become huge financial successes. The Onewheel (Fig. 11.1.1), the Dual-Wheel Hovercycle, and the Boardless Skateboard provide even newer, innovative rolling fun. Applying the physics of rolling can still lead to surprises and rewards. Our starting point in exploring that physics is to simplify Rolling Motion. Rolling as Translation and Rotation Combined Here we consider only objects that roll smoothly along a surface; that is, the objects roll without slipping or bouncing on the surface. Figure 11.1.2 shows how compli- cated smooth Rolling Motion can be: Although the center of the object moves in Figure 11.1.1 The Onewheel. Tania Whatley/Shutterstock.com 11.1 ROLLING AS TRANSLATION AND ROTATION COMBINED Learning Objectives After reading this module, you should be able to . . . 11.1.1 Identify that smooth rolling can be consid- ered as a combination of pure translation and pure rotation. 11.1.2 Apply the relationship between the center-of- mass speed and the angular speed of a body in smooth rolling.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
The physics and engineering of rolling have been around for so long that you might think no fresh ideas remain to be developed. However, skateboards and inline skates were invented and engineered fairly recently, to become huge financial successes. Street luge is now catching on, and the self-righting Segway (Fig. 11-1) may change the way people move around in large cities. Applying the physics of rolling can still lead to surprises and rewards. Our starting point in exploring that physics is to simplify Rolling Motion. Rolling as Translation and Rotation Combined Here we consider only objects that roll smoothly along a surface; that is, the objects roll without slipping or bouncing on the surface. Figure 11-2 shows how compli- cated smooth Rolling Motion can be: Although the center of the object moves in a straight line parallel to the surface, a point on the rim certainly does not. How- ever, we can study this motion by treating it as a combination of translation of the center of mass and rotation of the rest of the object around that center. 252 To see how we do this, pretend you are standing on a sidewalk watching the bicycle wheel of Fig. 11-3 as it rolls along a street. As shown, you see the center of mass O of the wheel move forward at constant speed v com . The point P on the street where the wheel makes contact with the street surface also moves forward at speed v com , so that P always remains directly below O. During a time interval t, you see both O and P move forward by a distance s. The bicycle rider sees the wheel rotate through an angle θ about the center of the wheel, with the point of the wheel that was touching the street at the beginning of t moving through arc length s. Equation 10-17 relates the arc length s to the rotation angle θ: s = θR, (11-1) where R is the radius of the wheel. The linear speed v com of the center of the wheel (the center of mass of this uniform wheel) is ds/dt. The angular speed of the wheel about its center is dθ/dt.
- David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
CHAPTER 11 Rolling, torque, and angular momentum 11.1 Rolling as translation and rotation combined LEARNING OBJECTIVES After reading this module, you should be able to: 11.1.1 identify that smooth rolling can be considered as a combination of pure translation and pure rotation 11.1.2 apply the relationship between the centre‐of‐mass speed and the angular speed of a body in smooth rolling. KEY IDEAS • For a wheel of radius R rolling smoothly, v com = R, where v com is the linear speed of the wheel’s centre of mass and is the angular speed of the wheel about its centre. • The wheel may also be viewed as rotating instantaneously about the point P of the ‘road’ that is in contact with the wheel. The angular speed of the wheel about this point is the same as the angular speed of the wheel about its centre. Why study physics? Analysing and predicting the destructive power arising from the motion of the atmosphere occurring during the cyclones battering Western Australia, the Northern Territory and Queensland, or the shifting of the Earth’s crust, producing the cataclysmic earthquakes of New Zealand, involves gaining a proficiency in understanding and applying the principles of rolling, torque and angular momentum. Rolling as translation and rotation combined FIGURE 11.1 The centre of mass O of a rolling wheel moves a distance s at velocity v com while the wheel rotates through angle . The point P at which the wheel makes contact with the surface over which the wheel rolls also moves a distance s. P P O O θ s v com s v com Here we consider only objects that roll smoothly along a surface; that is, the objects roll without slipping or bouncing on the surface. Our first concern is how various parts of such a rolling object move. The motion of any point other than the centre of mass appears to be very complicated.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
The physics and engineering of rolling have been around for so long that you might think no fresh ideas remain to be developed. However, skateboards and inline skates were invented and engineered fairly recently, to become huge financial successes. Street luge is now catching on, and the self-righting Segway (Fig. 11-1) may change the way people move around in large cities. Applying the physics of rolling can still lead to surprises and rewards. Our starting point in exploring that physics is to simplify Rolling Motion. Rolling as Translation and Rotation Combined Here we consider only objects that roll smoothly along a surface; that is, the objects roll without slipping or bouncing on the surface. Figure 11-2 shows how compli- cated smooth Rolling Motion can be: Although the center of the object moves in a straight line parallel to the surface, a point on the rim certainly does not. How- ever, we can study this motion by treating it as a combination of translation of the center of mass and rotation of the rest of the object around that center. 296 To see how we do this, pretend you are standing on a sidewalk watching the bicycle wheel of Fig. 11-3 as it rolls along a street. As shown, you see the center of mass O of the wheel move forward at constant speed v com . The point P on the street where the wheel makes contact with the street surface also moves forward at speed v com , so that P always remains directly below O. During a time interval t, you see both O and P move forward by a distance s. The bicycle rider sees the wheel rotate through an angle θ about the center of the wheel, with the point of the wheel that was touching the street at the beginning of t moving through arc length s. Equation 10-17 relates the arc length s to the rotation angle θ: s = θR, (11-1) where R is the radius of the wheel. The linear speed v com of the center of the wheel (the center of mass of this uniform wheel) is ds/dt. The angular speed of the wheel about its center is dθ/dt.
eBook - PDF- Richard L. Myers(Author)
- 2005(Publication Date)
- Greenwood(Publisher)
Q Rotational Motion Introduction In translational motion, an object moves from one point to another. Another type of motion occurs when an object moves in a curved path. This type of motion is called rotational motion. Examples of rota- tional motion are all around us. A spinning compact disc, the wheels of a car, and a blowing fan are a few common examples of rotational motion. As you sit reading this book, you are in constant rotational motion as the Earth spins on its axis. Rota- tional motion may accompany translational motion, as when a ball rolls down a ramp. It may also occur in the absence of trans- lational motion, as when a top spins on its axis. Rotation refers to the circular move- ment of an object about an axis that passes through the object. Another type of curved motion occurs when an object moves about an axis outside of it. This is called a revo- lution. Both rotational and revolutionary motion will be considered in this chapter, and the term rotational motion will be used as a generic description for curved motion. Many of the concepts for rotational motion parallel those for translational motion. The simplest type of rotational motion is uni- form circular motion. Uniform Circular Motion Circular is used as a general term to describe motion along a curved path and should not be interpreted to mean motion following a perfect circle; for instance, elliptical motion is classified as circular motion. Common examples of circular motion include the orbital motion of sat- ellites, amusement park rides, rounding a curve in a car, and twirling an object on the end of a string. Many of the concepts describing translational motion have anal- ogous counterparts for circular motion. These will be developed in this chapter. The concepts in this chapter apply to both rota- tional and revolutionary motion. Rotation involves circular motion around an axis that passes through the object itself, for example, a rotating top.
eBook - PDF- John Matolyak, Ajawad Haija(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
139 © 2010 Taylor & Francis Group, LLC Rotational Motion Translational motion and circular motion, uniform and nonuniform of an object, were discussed in Chapters 3 and 4. In this chapter, rotational motion of a point or extended object will be introduced. A force acting on an extended object creates a torque that rotates it about a fixed axis. A solid object of finite physical size is known as a rigid body. The axis of rotation could be about the center of mass of the rigid body or about other points where it is pivoted. In this chapter, Sections 8.2 through 8.7 discuss the kinematics of a rigid body, and Section 8.8 reviews the dynamics of the rigid body. In addition to drawing a parallel between Newton’s laws applied to point-like objects in linear motions and those applied to rotational motion of a rigid body, translational and rotational motions of a rigid body are discussed in detail. 8.1 ANGULAR KINEMATIC QUANTITIES Consider a point-like object moving in a circle of radius r (Figure 8.1). As the object moves from point P 1 at t 1 to point P 2 at t 2 , it sweeps through an arc Δ s that subtends an angle Δθ ( = θ 2 – θ 1 ) at the center, in a time interval Δ t = t 2 – t 1 . From geometry Δ s = r Δθ , (8.1) where Δθ is an angular displacement, measured in radians. To convert an angle expressed in degrees to radians, the following relation may be used: θ π θ rad deg deg 2 rad 360 = , (8.2a) and from radians to degrees, the conversion is θ π θ de g r ad 360 2 . = (8.2b) Dividing Equation 8.1 by Δ t gives the average linear velocity v ; that is, ∆ ∆ ∆ ∆ s t r t = θ or v r . = ω (8.3) For an object experiencing a constant linear acceleration, v v v 2 1 2 = + . 8 140 Essential Physics © 2010 Taylor & Francis Group, LLC Here, v 1 and v 2 are the instantaneous tangential, or linear, velocities of the object at points P 1 and P 2 , respectively.
- Raymond Serway, John Jewett(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Therefore, the translational speed of the center of mass for pure Rolling Motion is given by v CM 5 ds dt 5 R d u dt 5 Rv (10.28) where v is the angular speed of the cylinder. Equation 10.28 holds whenever a cyl- inder or sphere rolls without slipping and is the condition for pure Rolling Motion. The magnitude of the linear acceleration of the center of mass for pure Rolling Motion is a CM 5 dv CM dt 5 R dv dt 5 R a (10.29) where a is the angular acceleration of the cylinder. Imagine that you are moving along with a rolling object at speed v CM , staying in a frame of reference at rest with respect to the center of mass of the object. As you observe the object, you will see the object in pure rotation around its center of mass. Figure 10.25a shows the velocities of points at the top, center, and bottom of the object as observed by you. In addition to these velocities, every point on the object moves in the same direction with speed v CM relative to the surface on which it rolls. Figure 10.25b shows these velocities for a nonrotating object. In the refer- ence frame at rest with respect to the surface, the velocity of a given point on the object is the sum of the velocities shown in Figures 10.25a and 10.25b. Figure 10.25c shows the results of adding these velocities. Notice that the contact point between the surface and object in Figure 10.25c has a translational speed of zero. At this instant, the rolling object is moving in exactly the same way as if the surface were removed and the object were pivoted at point P and spun about an axis passing through P. We can express the total kinetic energy of this imagined spinning object as K 5 1 2 I P v 2 (10.30) where I P is the moment of inertia about a rotation axis through P. Because the motion of the imagined spinning object is the same at this instant as our actual rolling object, Equation 10.30 also gives the kinetic energy of the rolling object.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
316 CHAPTER 11 Rolling, Torque, and Angular Momentum Rolling Bodies For a wheel of radius R rolling smoothly, v com = R, (11.1.2) where v com is the linear speed of the wheel’s center of mass and is the angular speed of the wheel about its center. The wheel may also be viewed as rotating instantaneously about the point P of the “road” that is in contact with the wheel. The angular speed of the wheel about this point is the same as the angular speed of the wheel about its center. The rolling wheel has kinetic energy K = 1 _ 2 I com ω 2 + 1 _ 2 Mv com 2 , (11.2.3) where I com is the rotational inertia of the wheel about its center of mass and M is the mass of the wheel. If the wheel is being accelerated but is still rolling smoothly, the acceleration of the center of mass a → com is related to the angular acceleration about the center with a com = R. (11.2.4) If the wheel rolls smoothly down a ramp of angle θ, its accelera- tion along an x axis extending up the ramp is a com, x = − g sin θ ____________ 1 + I com / MR 2 . (11.2.8) Torque as a Vector In three dimensions, torque τ → is a vector quantity defined relative to a fixed point (usually an origin); it is τ → = r → × F → , (11.4.1) where F → is a force applied to a particle and r → is a position vector locating the particle relative to the fixed point. The magnitude of τ → is τ = rF sin ϕ = r F ⊥ = r ⊥ F, (11.4.2, 11.4.3, 11.4.4) where is the angle between F → and r → , F ⊥ is the component of F → perpendicular to r → , and r ⊥ is the moment arm of F → . The direc- tion of τ → is given by the right-hand rule. Angular Momentum of a Particle The angular momentum ℓ → of a particle with linear momentum p → , mass m, and linear veloc- ity v → is a vector quantity defined relative to a fixed point (usu- ally an origin) as ℓ → = r → × p → = m ( r → × v → ) .
eBook - PDFWorkshop Physics Activity Guide Module 2
Mechanics II
- Priscilla W. Laws, David P. Jackson, Brett J. Pearson(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
Name Section Date UNIT 12: ROTATIONAL MOTION Siegfried Haasch/Getty images Electricity is generated when the wind causes these large turbines to spin around in rotational motion. As can be observed by the blurring in the photograph, each point on the turbine moves at a different speed, with those farther from the rotation axis moving at faster speeds. But even though these points are all moving at different speeds, there is obviously something similar about their motion. In particular, each point on the turbine travels in a complete circle each time the wheel makes one full rotation. In this unit, we will learn how to describe such motion and how to approach solving problems that involve rotations. 390 WORKSHOP PHYSICS ACTIVITY GUIDE UNIT 12: ROTATIONAL MOTION OBJECTIVES 1. To understand the concepts of rotational position, velocity, and accelera- tion, as well as the associated kinematic equations. 2. To develop a definition for rotational inertia as a measure of the resistance to changes in rotational motion. 3. To understand torque and its relation to rotational acceleration and rota- tional inertia. 12.1 OVERVIEW We have studied the characteristics of an object undergoing translational motion, usually along a straight line, but also for objects traveling in circular motion or projectile motion. By “translational” motion we mean the motion of the object’s center of mass (or equivalently, the motion of a point mass), ignoring any rotation of the object. But in the real world, many objects will also rotate as they move. A rolling wheel, Earth spinning on its axis, and a hammer that tumbles when tossed are a few examples. However, our current formulation of Newton’s second law only applies to the center-of-mass (translational) motion of an object. In this unit we will define several new quantities and relationships to describe the rotational motion of rigid objects, objects that do not change their shape as they move.
eBook - PDF- Harold Josephs, Ronald Huston(Authors)
- 2002(Publication Date)
- CRC Press(Publisher)
( S could be a portion of a body upon which B rolls.) Let B and S be counterformal so that they are in contact at a single point. Let C be the point of B that is in contact with S . Rolling then occurs when: (4.11.1) FIGURE 4.11.1 A body B rolling on a surface S . S L S O S C S C y z z y z y a a OL n n n n n n * . . . . . . = + × × ( ) = − + − ( ) × − ( ) × + ( ) [ ] = − ω ω 19 36 0 88 0 88 0 433 0 25 19 7 ft sec 2 2 2 0 88 1 571 2 71 2 765 S C C L z y z x ω × = − ( ) × − + ( ) = − V n n n n . . . . ft sec 2 S L x y z a n n n = − − − 2 765 36 79 9 87 . . . ft sec 2 S C V = 0 B P p C Kinematics of a Rigid Body 107 Rolling Motion is governed by the magnitude and direction of the angular velocity of B in S . Let n be a unit vector normal to S at C. Then B has pure rolling in S if the angular velocity of B in S is perpendicular to n . That is, (4.11.2) B is pivoting in S if S ω B is parallel to n . That is, (4.11.3) B is at rest relative to S if S ω B is zero: (4.11.4) Finally, B has general rolling in S if B is neither at rest nor pivoting or has pure rolling in S . (Pure rolling is desired in machine elements to reduce the wear of the rolling surfaces.) Consider again Figure 4.11.1. Let P be an arbitrary point of B . Because C is fixed in B and because S V C is zero, Eqs. (3.4.6), (4.5.2), and (4.9.4) show that the velocity of P in S is simply: (4.11.5) where p is the position vector locating P relative to C . The acceleration of P in S may be obtained by differentiating in Eq. (4.11.5). 4.12 The Rolling Disk and Rolling Wheel As an illustration of these ideas, consider a uniform circular disk D rolling on a horizontal flat surface S as depicted in Figure 4.12.1 (see References 4.1 to 4.4). Let C be the contact point between D and S , and let G be the center of D . Let axes X , Y , and Z form a Cartesian reference frame fixed relative to S with the Z -axis being normal to S . Let N 1 , N 2 , and N 3 be unit vectors parallel to X , Y , and Z .
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