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Chapter 11
Rolling, Torque, and Angular Momentum
In this chapter we will cover the following topics:
-Rolling of circular objects and its relationship with friction
-Redefinition of torque as a vector to describe rotational problems that are more complicated than the rotation of a rigid body about a fixed axis
-Angular momentum of single particles and systems of particles
-Newton’s second law for rotational motion
-Conservation of angular momentum
-Applications of the conservation of angular momentum
(11-1)
t1 = 0
t2 = t
Rolling as Translation and Rotation Combined
Consider an object with circular cross section that rolls along a surface without slipping. This motion, though common, is complicated. We can simplify its study by treating it as a combination of translation of the center of mass and rotation of the object about the center of mass.
Consider the two snapshots of a rolling bicycle wheel shown in the figure.
An observer stationary with the ground will see the center of mass O of the wheel move forward with a speed vcom . The point P at which the wheel makes contact with the road also moves with the same speed. During the time interval t between ds the two snapshots both O and P cover a distance s, vcom =
(eq. 1). During t dt the bicycle rider sees the wheel rotate by an angle θ about O so that ds dθ s = Rθ →
=R
=ω (eq. 2). If we combine equation 1 with equation 2 dt dt we get the condition for rolling without slipping: vcom = Rω
(11-2)
vcom = Rω
We have seen that rolling is a combination of purely translational motion with speed vcom and a purely rotational motion about the center of mass vcom . The velocity of each point is the vector sum
R
of the velocities of the two motions. For the translational motion the r velocity vector is the same for every point (vcom ,see fig. b). The rotational with angular velocity ω =
velocity varies from point to point. Its magnitude is equal to ω r where r is
Rolling, Torque, and Angular Momentum
In this chapter we will cover the following topics:
-Rolling of circular objects and its relationship with friction
-Redefinition of torque as a vector to describe rotational problems that are more complicated than the rotation of a rigid body about a fixed axis
-Angular momentum of single particles and systems of particles
-Newton’s second law for rotational motion
-Conservation of angular momentum
-Applications of the conservation of angular momentum
(11-1)
t1 = 0
t2 = t
Rolling as Translation and Rotation Combined
Consider an object with circular cross section that rolls along a surface without slipping. This motion, though common, is complicated. We can simplify its study by treating it as a combination of translation of the center of mass and rotation of the object about the center of mass.
Consider the two snapshots of a rolling bicycle wheel shown in the figure.
An observer stationary with the ground will see the center of mass O of the wheel move forward with a speed vcom . The point P at which the wheel makes contact with the road also moves with the same speed. During the time interval t between ds the two snapshots both O and P cover a distance s, vcom =
(eq. 1). During t dt the bicycle rider sees the wheel rotate by an angle θ about O so that ds dθ s = Rθ →
=R
=ω (eq. 2). If we combine equation 1 with equation 2 dt dt we get the condition for rolling without slipping: vcom = Rω
(11-2)
vcom = Rω
We have seen that rolling is a combination of purely translational motion with speed vcom and a purely rotational motion about the center of mass vcom . The velocity of each point is the vector sum
R
of the velocities of the two motions. For the translational motion the r velocity vector is the same for every point (vcom ,see fig. b). The rotational with angular velocity ω =
velocity varies from point to point. Its magnitude is equal to ω r where r is