PHY 001 (Physics I)

Lecture 6

Instructor: Dr. Mohamed Fouad Salem

mohamed.salem@gmail.com

TextbookUniversity Physics, 12th edition,

Young and Freedman

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Course Material Website

http://meryesk.wordpress.com/phy001/

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Chapter 9

Rotation of Rigid Bodies

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• You are asked to design an airplane propeller to turn at 2400 rpm. The forward airspeed of the plane to be 75.0 m/s, and the speed of the tips of the propeller blades through the air must no exceed 270 m/s.

Relating Linear and Angular Kinematics(Example 2)

air must no exceed 270 m/s. (a)What is the maximum radius the propeller

can have?(b) With this radius, what is the acceleration of

the propeller tip?

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Relating Linear and Angular Kinematics(Example 2)

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• (a)Then

Relating Linear and Angular Kinematics(Example 2 Solution)

• The centripetal acceleration is

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• How are the angular speeds of the two bicycle sprockets in the shown figure related to the number of teeth on each sprocket?

Relating Linear and Angular Kinematics(Example 3)

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• The linear speed is constant for both sprockets, since the chain does not stretch or slip, hence

• The teeth must be equally spaced on the circumferences of both sprockets for the chain to

Relating Linear and Angular Kinematics(Example 3 Solution)

• The teeth must be equally spaced on the circumferences of both sprockets for the chain to mesh properly with both

• So

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9-4 Rotational Kinetic Energy A set of masses mi uniformly rotating with angular velocity ω about some fixed axis A possesses a kinetic energy defined by

where ri is the distance from the ith mass to the rotation axis.For such a set of mass, or for a continuous body, we

K = 12 miv i

2 = 12 mi

i

∑ ri2ωωωω 2

i

∑

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For such a set of mass, or for a continuous body, we define the moment of inertia I about the specified axis Aas

Then the rotational kinetic energy can be written as

I = miri2

i

∑

K = 12 Iω

2

Moment of Inertia

• The moment of inertia of a set of particles is

I = m1r12 + m2r22 + … = Σmiri2

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• The rotational kinetic energy of a rigid bodyhaving a moment of inertia I is

K = 1/2 Iω2

Moment of Inertia The greater the moment of inertia of a rigid body, the more difficult to make it rotate if it is at rest or to stop it if it started rotating

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Moment of Inertia for different rotation axes(Example)

An engineer is designing a machine part consisting of three heavydisks linked by lightweight struts as shown(a) What it is the moment of inertia of this body about an axis

through the center of disk A, perpendicular to the plane of thediagram?

(b) What it is the moment of inertia about an axis through the center of disks B and C?

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center of disks B and C?(c) If the body rotates about

an axis through A as in (a) with angular speedω = 4.0 rad/s, what it is the kinetic energy?

Moment of Inertia for different rotation axes(Example solution)

(a)

(b)

(c)

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Moments of inertia of some common bodies

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Rotational Energy (Example 1)

• We wrap a light, nonstretching cable around a solid cylinder of mass 50 kg and diameter 0.120 m, which rotates in frictionless bearings about a stationary axis. We pull the free end of the cable with a constant 9.0 N force for a distance of 2.0 m; it turns the cylinder as it unwinds without slipping. The cylinder is initially at rest.

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cylinder is initially at rest. Find its final angularspeed and the finalspeed of the cable.

The work done on the cylinder is:

The moment of inertia is:

Rotational Energy (Example 1 solution)

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Conservation of energygives:

Rotational Energy (Example 1 solution continuation)

The final tangential speed of the cylinder, and hence

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The final tangential speed of the cylinder, and hence the final speed of the cable is:

Rotational Energy (Example 2) We wrap a light, nonstretching cable around a solid cylinder with mass M and radius R. The cylinder rotates with negligible friction about a stationary horizontal axis. We tie the free end of the cable to a block of mass m and release the block from rest at a distance h above the floor. As the block falls,the cable unwinds without stretchingor slipping. Find expressions

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for the speed of the falling blockand the angular speed of thecylinder as the block strikesthe floor.

Rotational Energy (Example 2 solution)

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Rotational Energy (Example 2 solution continuation)

Solving for the linear velocity gives:

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Gravitational potential energy of an extended body

• In the previous example if the cable were to have considerable mass not negligible as assumed, we need to calculate gravitational potential energy for it.

• The gravitational potential energy of an

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• The gravitational potential energy of an extended body is the same as if all the mass were concentrated at its center of mass:

Ugrav = Mgycm

• Where ycm is the y-coordinate of the center of mass.

Next Time

• Section 9-5 Continued

• Section 9-6

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Assignment # 4

• Section 9-434, 35, 39, 41, 47, 49 and 53

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