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)

t 1 = 0t 2 = t (11-2)

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A B vAvA vBvB vTvT vOvO (11-4)

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a com (11-7)

a com (11-8)

Example: A uniform cylinder rolls down a ramp inclined at an angle of θ to the horizontal. What is the linear acceleration of the cylinder at the bottom of the ramp? Remember that: The friction force is used to rotate the object.

Example: Consider a solid cylinder of radius R that rolls without slipping down an incline from some initial height h. The linear velocity of the cylinder at the bottom of the incline is vcm and the angular velocity is ω.

We can also solve for angular velocity using the equation

Example: A bowling ball has a mass of 4.0 kg, a moment of inertia of 1.6×10**(−2) kg ·m2 and a radius of 0.10 m. If it rolls down the lane without slipping at a linear speed of 4.0 m/s, what is its total energy?

y a com (11-9)

B (11-10)

Torque Revisited Example: (T72_Q17.) A force is applied to an object that is pivoted about a fixed axis aligned along the z-axis. If the force is applied at the point of coordinates (4.0, 5.0, 0.0) m, what is the applied torque (in N.m) about the z axis?

Example: (T71_Q19.). At an instant, a particle of mass 2.0 kg has a position of and acceleration of. What is the net torque on the particle at this instant about the point having the position vector: ?

B (11-11)

Example: (T052 Q#18) A stone attached to a string is whirled at 3.0 rev/s around a horizontal circle of radius 0.75 m. The mass of the stone is 0.15 kg. The magnitude of the angular momentum of the stone relative to the center of the circle is:

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O m1m1 m3m3 m2m2 ℓ1ℓ1 ℓ2ℓ2 ℓ3ℓ3 x y z (11-13)

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y-axis (11-17)

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