Planar Kinetics of a Rigid BodyWork and Energy

Outline

• Kinetic energy

• Work of a force

• Work of a couple

• Principle of work and energy

• Conservation of energy

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Kinetic Energy

• At arbitrary ith particle of the body

• Kinetic energy of the entire body

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2

2

1ii vdmTdT

m

ivdmT 2

2

1

Kinetic Energy

• If the body has an angular velocity ω, then

• The square of the magnitude of vi is thus

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ji

jikji

vvv

])[(])[(

)()()(

/

xvyv

yxvv

yPxP

yPxP

PiPi

222

222222

222

)(2)(2

)(2)()(2)(

])[(])[(

rxvyvv

xxvvyyvv

xvyvv

yPxPP

yPyPxPxP

yPxPiii

vv

Kinetic Energy

• Substituting into the equation of kinetic equation yields

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mm

yPm

xPPm

dmrdmxvdmyvvdmT 222

2

1)()(

2

1

Translational kinetic energy of entire body @ point P

mydmmy m

xdmmx

Location of CG with respect to point P

Rotational kinetic energy of entire body @ point P

Kinetic Energy

• Special case, P = mass center G then

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22

2

1)()(

2

1 PyPxPP ImxvmyvmvT

0 xy

22

2

1

2

1GG ImvT

Kinetic Energy

• When a rigid body of mass m is subjected to either rectilinear or curvilinear translation, the kinetic energy due to rotation is zero.

• The kinetic energy of the body is therefore

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Translation

2

2

1GmvT

Kinetic Energy

• When the rigid body is rotating about a fixed axis passing through point O, the body has both translational and rotational kinetic energy as defined by

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Rotation About a Fixed Axis

22

2

1

2

1GG ImvT 2

2

1OIT

Parallel-axis Theorem

Kinetic Energy

• When the rigid body is subjected to general plane motion, it has an angular velocity ω and its mass center has a velocity vG

• The kinetic energy is defined by

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General Plane Motion

22

2

1

2

1GG ImvT

Example 1

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The system of three elements shown consists of a 6-kg block B, a 10-kg disk D and a 12-kg cylinder C. If no slipping occurs, determine the total kinetic energy of the system at the instant shown.

Work of a force

• If an external force F acts on a rigid body, the work done by the force when it moves along the path s,

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s

F dsFU cosrF

Work of a Variable Force

Work of a force

• The weight of a body does work only when the body’s center of mass G undergoes a vertical displacement Δy.

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Work of a Weight

yWUw

Work of a force

• If a linear elastic spring is attached to a body, the spring force Fs = ks acting on the bodydoes work when the spring either stretches or compresses from s1 to a further position s2.

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Work of a Spring Force

2

122

2

1

2

1ksksUs

12 ss

Work of a force

• These forces can either act at fixed points on the body, or they can have a direction perpendicular to their displacement.

• Example includes the weight of a body when the center of gravity of the body moves in a horizontal plane.

• A rolling resistance force Fr acting on a round body as it rolls without slipping over a rough surface also does no work.

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Forces That Do Not Work

Work of a couple

• Consider the body as shown, which is subjected to a couple moment M = Fr.

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Work of a couple

• When the body translates, such that the component of displacement along the line of action of the forces is dst

• Clearly the positive work of one force cancels the negative work of the other

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Work of a couple

• If the body undergoes a differential rotation dθ about an axis which is perpendicular to the plane of the couple and intersects the plane at point O, then each force undergoes a displacement dsθ = (r/2) dθ in the direction of the force.

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Work of a couple

• The total work done during dθ is

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dM

dFrdr

Fdr

FdUM

)(

22

)( 12

2

1

MdMUM

Constant magnitude

Example 2

The bar shown has a mass of 10-kg and is subjected to a couple moment of M = 50 N.m and a force of P = 80 N, which is always applied perpendicular to the end of the bar. Also, the spring has an unstretchedlength of 0.5 m and remains in the vertical position due to the roller guide at B. determine the total work done by all the forces acting on the bar when it has rotated downward from θ = 0 to θ° = 90°.

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Principle of Work and Energy

• T1 = Initial translational and rotational K.E.

• ΣU1-2 = Work done by external forces and couple moments

• T2 = Final translational and rotational K.E.

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2211 TUT

All internal forces produce zero work

Example 3

The 10-kg rod is constrained so that its ends move along the grooved slots. The rod is initially at rest when θ = 0°. If the slider block at B is acted upon by a horizontal force P = 50 N, determine the angular velocity of the rod at the instant θ = 45°.

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Problem 18-17

The 4-kg slender rod is subjected to the force and couple moment. When the rod is in the position shown it has a angular velocity ω1 = 6 rad/s. Determine its angular velocity at the instant it has rotated 360⁰. The force is always applied perpendicular to the axis of the rod and motion occurs in the vertical plane.

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Conservation of Energy

• When a force system acting on a rigid consists only of conservative forces, the conservation of energy theorem may be used to solve a problem which otherwise would be solved using the principle of work and energy.

• This theorem is easier to apply since the work of a conservative force is independent of the path and depends only on the initial and final positions of the body.

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Conservation of Energy

• Since the total weight of a body can be considered concentrated at it center of gravity, the gravitational potential energy of the body is determined by knowing the height of the body’s CG above or below a horizontal datum.

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Gravitational Potential Energy

Conservation of Energy

• Measuring yG as positive upward, the gravitational potential energy of the body is thus,

• Here the potential energy is positive when yG

is positive, since the weight has the ability to do positive work when the body is moved back to the datum.

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Gg yWV

Conservation of Energy

• The force developed by an elastic spring is conservative force.

• The elastic potential energy which a spring imparts to an attached body when the spring is elongated or compressed from an initial undeformed position (s = 0) to a final position s, is

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Elastic Potential Energy

2

2

1ksVe

Conservation of Energy

• In general, if a body is subjected to both gravitational and elastic forces, the total potential energy is expressed as a potential function V represented as the algebraic sum

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Potential Energy

eg VVV

Conservation of Energy

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221211 )) TUUTnonconscons

Principle of Work and Energy

2121 ) VVUcons

222111 ) VTUVTnoncons

2211 VTVT 0)21 nonconsU

Problem 18-46

The 40-kg cylinder is attached to the 5-kg slender rod which is pinned at point A. At the instant θ=30⁰ the rod has an angular velocity of ωo= 1 rad/s as shown. Determine the angle θ to which the rod swings before it momentarily stops.

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