Equilibrium and Elasticity - University of Texas at Dallasdavid.lary/CM/Classical_Mechanics/... ·...

Equilibrium and Elasticity

• Many bodies, such as bridges, aqueducts, and ladders, are designed so they do not accelerate.

• Real materials are not truly rigid. They are elastic and do deform to some extent.

• We consider stress and strain to understand the deformation of real bodies.

Tuesday, October 18, 11

Acts like a spring

QuizHooke’s law describes the force of

A. gravity.B. a spring.C. collisions.D. tension.E. none of the above.

AnswerHooke’s law describes the force of

A. gravity.B. a spring.C. collisions.D. tension.E. none of the above.

Checking UnderstandingWhich spring has the largest spring constant?

Which spring has the largest spring constant?

Answer

Checking UnderstandingThe same spring is stretched or compressed as shown below. In which case does the force exerted by the spring have the largest magnitude?

E. Not enough information to tell.

The same spring is stretched or compressed as shown below. In which case does the force exerted by the spring have the largest magnitude?

Answer

Torque and Static EquilibriumFor an extended object to be in equilibrium, the net force and the net torque must be zero.

Choosing the Pivot Point

Conditions for equilibrium• First condition: The sum of all the

forces is equal to zero:

ΣFx = 0 ΣFy = 0 ΣFz = 0

• Second condition: The sum of all torques about any given point is equal to zero.

Center of gravity

• We can treat a body’s weight as though it all acts at a single point—the center of gravity.

• If we can ignore the variation of gravity with altitude, the center of gravity is the same as the center of mass.

Solving Static Equilibrium Problems

Checking UnderstandingWhat does the scale read?

A. 500 NB. 1000 NC. 2000 ND. 4000 N

AnswerWhat does the scale read?

A. 500 NB. 1000 NC. 2000 ND. 4000 N

Balance

For an object to balance, its center of gravity must reside over its base of support. That way gravity does not exert a torque.

Base of support

Gravity acts at the center of gravity.

Line of action

This force exerts no torque about her toes.

Stability of a Car

TiptoeingWhy can’t you stand on tiptoes if your toes are against a wall?

Center of gravity has to be over toes – the base of support – to balance. That requires shifting your body slightly forward. But you can’t shift your body forward if your toes are against the wall.

Tensile and compressive stress and strain• Tensile stress = F⊥ /A and tensile strain = Δl/l0.

• Compressive stress and compressive strain are defined in a similar way.

• Young’s modulus is tensile stress divided by tensile strain, and is given by Y = (F⊥/A)(l0/Δl).

Some values of elastic moduli

Tensile stress and strain

• In many cases, a body can experience both tensile and compressive stress at the same time

Bulk stress and strain

• Pressure in a fluid is force per unit area: p = F⊥/A.

• Bulk stress is pressure change Δp and bulk strain is fractional volume change ΔV/V0.

• Bulk modulus is bulk stress divided by bulk strain and is given by B = –Δp/(ΔV/V0).

Sheer stress and strain

• Sheer stress is F||/A and sheer strain is x/h

• Sheer modulus is sheer stress divided by sheer strain, and is given by S = (F||/A)(h/x).

Elasticity and plasticity

• Hooke’s law applies up to point a

• Table 11.3 shows some approximate breaking stresses.

Which of the following situations satisfies both the first condition for equilibrium (net force = 0) and the second condition for equilibrium (net torque = 0)?

A. an automobile crankshaft turning at an increasing angular speed in the engine of a parked car

B. a seagull gliding at a constant angle below the horizontal and at a constant speed

C. a thrown baseball that does not rotate as it sails through the air

D. more than one of the above

E. none of the above

Which of the following situations satisfies both the first condition for equilibrium (net force = 0) and the second condition for equilibrium (net torque = 0)?

A. an automobile crankshaft turning at an increasing angular speed in the engine of a parked car

B. a seagull gliding at a constant angle below the horizontal and at a constant speed

C. a thrown baseball that does not rotate as it sails through the air

D. more than one of the above

A rock is attached to the left end of a uniform meter stick that has the same mass as the rock. How far from the left end of the stick should the triangular object be placed so that the combination of meter stick and rock is in balance?

A.less than 0.25 m

B. 0.25 m

C. between 0.25 m and 0.50 m

D. 0.50 m

E. more than 0.50 m

A rock is attached to the left end of a uniform meter stick that has the same mass as the rock. How far from the left end of the stick should the triangular object be placed so that the combination of meter stick and rock is in balance?

A.less than 0.25 m

B. 0.25 m

C. between 0.25 m and 0.50 m

D. 0.50 m

E. more than 0.50 m

L0.5-L

L × m = (0.5 − L) × mL = (0.5 − L)2L = 0.5L = 0.25

A. T = w sin θ

B. T = w cos θ

C. T = w/(sin θ)

D. T = w/(cos θ)

Q11.3A metal advertising sign (weight w) is suspended from the end of a massless rod of length L. The rod is supported at one end by a hinge at point P and at the other end by a cable at an angle θ from the horizontal.

What is the tension in the cable?

A11.3A metal advertising sign (weight w) is suspended from the end of a massless rod of length L. The rod is supported at one end by a hinge at point P and at the other end by a cable at an angle θ from the horizontal.

What is the tension in the cable?A. T = w sin θ

B. T = w cos θ

C. T = w/(sin θ)

D. T = w/(cos θ)

E. none of the aboveTuesday, October 18, 11

A. more stress and more strain.

B. the same stress and more strain.

C. the same stress and less strain.

D. less stress and less strain.

E. the same stress and the same strain.

Q11.5Two rods are made of the same kind of steel and have the same diameter. FF length 2L

FF length L

A force of magnitude F is applied to the end of each rod. Compared to the rod of length L, the rod of length 2L has

A11.5Two rods are made of the same kind of steel and have the same diameter. FF length 2L

FF length L

Q11.6Two rods are made of the same kind of steel. The longer rod has a greater diameter.

FF length 2L

FF length L

A11.6Two rods are made of the same kind of steel. The longer rod has a greater diameter.

FF length 2L

FF length L

Energy and WorkTuesday, October 18, 11

What Energy Transfer is Occurring Here?

How do we transfer energy?

Power: The rate at which energy is transformed from one kind to

another.Tuesday, October 18, 11

Quiz 1. If a system is isolated, the total energy of the system

A. increases constantly.B. decreases constantly.C. is constant.D. depends on work into the system.E. depends on work out of the system.

Answer1. If a system is isolated, the total energy of the system

A. increases constantly.B. decreases constantly.C. is constant.D. depends on work into the system.E. depends on work out of the system.

Quiz • If you raise an object to a greater height, you are increasing

A. kinetic energy. B. heat. C. potential energy.D. chemical energy.E. thermal energy.

Answer• If you raise an object to a greater height, you are increasing

A. kinetic energy. B. heat. C. potential energy.D. chemical energy.E. thermal energy.

Forms of EnergyMechanical Energy

Thermal Energy

Other forms include

The Basic Energy Model

Energy Transformations

Kinetic energy K = energy of motion

Potential energy U = energy of position

Thermal energy Eth = energy associated with temperature

System energy E = K + U + Eth + Echem + ...

Energy can be transformed within the system without loss.

Energy is a property of a system.

Energy Transformations

Kinetic energy K = energy of motion

Potential energy U = energy of position

Thermal energy Eth = energy associated with temperature

System energy E = K + U + Eth + Echem + ...

Energy can be transformed within the system without loss.

Energy is a property of a system.

Some Energy Transformations

Echem → Ug

Echem → Ug K → Eth

Echem → Ug

Echem → Eth

Echem → Ug

Echem → Eth

Echem → Ug Us → K → Ug

Echem → Eth

Checking UnderstandingA child is on a playground swing, motionless at the highest point of his arc. As he swings back down to the lowest point of his motion, what energy transformation is taking place?

A. K → Ug

B. Ug→ Eth C. Us → Ug

D. Ug → K E. K → Eth

AnswerA child is on a playground swing, motionless at the highest point of his arc. As he swings back down to the lowest point of his motion, what energy transformation is taking place?

A. K → Ug

B. Ug→ Eth C. Us → Ug

D. Ug → K E. K → Eth

Energy Transfers

These change the energy of the system.

Interactions with the environment.

Work is the mechanical transfer of energy to or from a system via pushes and pulls.

Energy Transfers: Work

W→ K

W→ K W→ Eth

W→ Us

Elastic Potential EnergyTuesday, October 18, 11

In part (a) of the figure, an air track cart attached to a spring rests on the track at the position xequilibrium and the spring is relaxed. In (b), the cart is pulled to the position xstart and released. It then oscillates about xequilibrium.Which graph correctly represents the potential energy of the spring as a function of the position of the cart?

Answer: 3.The cart starts at xstart with no kinetic energy, and so the spring’spotential energy is a maximum. Once released, the cart accelerates to theright and its kinetic energy increases as the potential energy of the springis converted into kinetic energy of the cart. As the cart passes the equilibriumposition, its kinetic energy is a maximum and so the spring’s potentialenergy is a minimum. Once to the right of xequilibrium, the cart starts tocompress the spring and it slows down as its kinetic energy is convertedback to potential energy of the recompressed spring. At the rightmost pointit reaches, the cart reverses its direction of travel. At that instant, it has nokinetic energy and the spring again has maximum potential energy.

• We’ve studied how Newton’s Second Law allows us to calculate an acceleration from a force but what if the force changes during its application? We must be able to account for things like an archer’s bow.

• We must look at action–reaction pairs that are not immediately obvious (like the shotgun expelling the pellets with expanding gas but having the expanding gas do work on the shotgun at the same

The Law of Conservation of Energy

The Work-Energy Equation

Work Done by Force at an Angle to Displacement

Use the parallel component if the force acts at an angle

How can it be such a great “workout” with no work?When positive and negative work cancel, the net work is zero even though muscles are exercising.

The work-energy theoremWork done on an object can change its motion and energy.

We can compare the kinetic energy of different bodies

Changes in the energy of a moving body under the influence of an applied force change differently depending on the direction of application.

The Basic EquationKf + Uf + ΔEth = Ki + Ui

A few things to note:• Work can be positive (work in) or negative (work out)• We are, for now, ignoring heat.• Thermal energy is…special. When energy changes to thermal energy, this change is irreversible.

Stepwise solution of work done by several forces

Energy Equations

Each of the boxes, with masses noted, is pulled for10 m across a level, frictionless floor by the noted force. Which box experiences the largest change in kinetic energy?

Checking Understanding

AnswerEach of the boxes, with masses noted, is pulled for10 m across a level, frictionless floor by the noted force. Which box experiences the largest change in kinetic energy?

Each of the boxes, with masses noted, is pulled for10 m across a level, frictionless floor by the noted force. Which box experiences the smallest change in kinetic energy?

Answer Each of the boxes, with masses noted, is pulled for10 m across a level, frictionless floor by the noted force. Which box experiences the smallest change in kinetic energy?

Example ProblemA 200 g block on a frictionless surface is pushed against a spring with spring constant 500 N/m, compressing the spring by 2.0 cm. When the block is released, at what speed does it shoot away from the spring?

Use :F = ma(Fsp )x = −kΔx

vx2 = v0x

2 + 2ax (x − x0 )

Work and energy with varying forcesPerhaps the best example is driving a car, alternating your attention between the gas and the brake.

The effect is a variable positive or negative force of various magnitude along a straight line.

The stretch of a spring and the force that caused it

The force applied to an ideal spring will be proportional to its stretch.

The graph of force on the y axis versus stretch on the x axis will yield a slope of k, the spring constant.

Stepping on a scaleWhether you like the result or not, stepping on a scale is an excellent example of applied force and the work being done to compress that spring.

Motion with a varying force

Motion on a curved pathIf you watch a child on a swing set, you can also consider the motion of a particle along a curved path.

Watt about power?Once work is calculated, dividing by the time that passed determines power.

The pun is credit to James Watt. (You will see that scientists of that era often were privileged to leave their names on the topic of their efforts.)

Also note the popular culture power unit of horsepower.

The energy you use may be noted from the meter the electric company probably installed to measure your consumption of energy in kilowatt-hours.

Power• Same mass...• Both reach 60 mph...

Same final kinetic energy, butdifferent times mean different powers.

An example you might do if the elevator is out

It’s interesting how a lighter stair climber and heavier stair climber can expend the same power by using different amounts of time.

Elastic Collisions

Four toy cars accelerate from rest to their top speed in a certain amount of time. The masses of the cars, the final speeds, and the time to reach this speed are noted in the table. Which car has the greatest power?

Car Mass (g) Speed (m/s) Time (s)

A 100 3 2

B 200 2 2

C 200 2 3

D 300 2 3

E 300 1 4

AnswerFour toy cars accelerate from rest to their top speed in a certain amount of time. The masses of the cars, the final speeds, and the time to reach this speed are noted in the table. Which car has the greatest power?

A 100 3 2

B 200 2 2

C 200 2 3

D 300 2 3

E 300 1 4

Four toy cars accelerate from rest to their top speed in a certain amount of time. The masses of the cars, the final speeds, and the time to reach this speed are noted in the table. Which car has the smallest power?

A 100 3 2

B 200 2 2

C 200 2 3

D 300 2 3

E 300 1 4

AnswerFour toy cars accelerate from rest to their top speed in a certain amount of time. The masses of the cars, the final speeds, and the time to reach this speed are noted in the table. Which car has the smallest power?

A 100 3 2

B 200 2 2

C 200 2 3

D 300 2 3

E 300 1 4

Summary

Additional QuestionsEach of the 1.0 kg boxes starts at rest and is then pushed for 2.0 m across a level, frictionless floor by a rope with the noted force. Which box has the highest final speed?

AnswerEach of the 1.0 kg boxes starts at rest and is then pushed for 2.0 m across a level, frictionless floor by a rope with the noted force. Which box has the highest final speed?

Equilibrium and Elasticity - University of Texas at Dallasdavid.lary/CM/Classical_Mechanics/... ·...

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