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Plane Motion of Rigid

Bodies: Energy Methods

Reference: Beer, Ferdinand P. et al, Vector Mechanics for Engineers : Dynamics, 8th Edition, Mc GrawHill

Hibbeler R.C., Engineering Mechanics: Dynamics, 11th Edition, Prentice Hall (Chapter 18)

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Work of Forces Acting on a Rigid Body

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Work of Forces Acting on a Rigid Body

Forces acting on rigid bodies which do no work:

• Forces applied to fixed points:

- reactions at a frictionless pin when the supported body

rotates about the pin.

• Forces acting in a direction perpendicular to the displacement

of their point of application:

- reaction at a frictionless surface to a body moving along

the surface

- weight of a body when its center of gravity moves

horizontally

• Friction force at the point of contact of a body rolling without

sliding on a fixed surface.

0 dtvFdsFdU cC

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Kinetic Energy of a Rigid Body in Plane Motion

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Conservation of Energy

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Power • Power = rate at which work is done

• For a body acted upon by force and moving with velocity , F

v

vFdt

dU Power

• For a rigid body rotating with an angular velocity and acted

upon by a couple of moment parallel to the axis of rotation,

M

Mdt

dM

dt

dUPower

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Sample Problem 17.1

For the drum and flywheel,

The bearing friction is equivalent to a

couple of At the instant shown,

the block is moving downward at 6 m /s

2kgm5.10I

mN60

Determine the velocity of the block after it

has moved 4 m downward.

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Sample Problem 17.1

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Sample Problem 17.2

mm80kg3

mm200kg10

BB

AA

km

km

The system is at rest when a moment

of is applied to gear B.

Neglecting friction, a) determine the

number of revolutions of gear B before

its angular velocity reaches 600 rpm,

and b) tangential force exerted by gear

B on gear A.

mN6 M

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Sample Problem 17.2

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Sample Problem 17.3

A sphere, cylinder, and hoop, each

having the same mass and radius, are

released from rest on an incline.

Determine the velocity of each body

after it has rolled through a distance

corresponding to a change of elevation h.

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Sample Problem 17.3

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Sample Problem 17.4

A 30-N slender rod pivots about the

point O. The other end is pressed

against a spring (k = 1800 N/m) until

the spring is compressed 30 cm and the

rod is in a horizontal position.

If the rod is released from this position,

determine its angular velocity and the

reaction at the pivot as the rod passes

through a vertical position.

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Sample Problem 17.4

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Sample Problem 17.5

Each of the two slender rods has a

mass of 6 kg. The system is released

from rest with b = 60o.

Determine a) the angular velocity of

rod AB when b = 20o, and b) the

velocity of the point D at the same

instant.

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Sample Problem 17.5

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Sample Problem 17.5

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Find: The angular velocity of rod AB at

= 0° if the rod is released from

rest when = 30°.

EXAMPLE 1

Given: The rod AB has a mass of 10 kg.

Piston B is attached to a spring of

constant k = 800 N/m. The spring

is un-stretched when = 0°.

Neglect the mass of the pistons.

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EXAMPLE 1 (continued)

Initial Position Final Position

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EXAMPLE 1 (continued)

Initial Position Final Position

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EXAMPLE 2

Given: The weight of the disk is 30 N and

its kG equals 0.6 m. The spring has

a stiffness of 2 N/m and an

unstretched length of 1 m.

Find: The velocity at the instant G

moves 3 m to the left. The disk is

released from rest in the position

shown and rolls without slipping.

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EXAMPLE 2 (continued)

(1) Initial Position

(2) Final Position

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EXAMPLE 2 (continued)

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