Chapter 12 Lecture

Slide 12-1

Essential University Physics Richard Wolfson

2nd Edition

© 2012 Pearson Education, Inc.

Static Equilibrium 靜止的平衡狀態

• Pulley: 滑車, 滑輪

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In this lecture you’ll learn簡介

•  To describe the conditions necessary for static equilibrium 有何條件？

•  To calculate forces and torques needed to ensure that a system is in static equilibrium 要得到滿足條件的力與力矩

•  To determine whether or not an equilibrium is stable是否穩定？

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Conditions for Static Equilibrium條件I •  A system in static equilibrium undergoes neither rotational

nor linear acceleration.沒有加速度, 沒有角加速度 –  If it’s at rest, it remains at rest.

如果是靜止的, 就保持靜止

•  The conditions for static equilibrium are –  No net force: 合力為零 –  No net torque: 和力矩為零

•  Torques can be evaluated about any convenient point:

計算的時候,力矩可以用一個方便的轉點來計算.

Fi∑ =

0

τ i∑ =

ri×Fi∑ =

0

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Conditions for Static Equilibrium條件 II Example: A drawbridge of length L and the forces acting on it. Choosing the pivot point at the lower left leaves

two forces that contribute torques: 例如:求把橋打開所需要的力, 選左下點為支點, 就只有2力的力矩不為零.

Torque due to tension T : τT = LT sinθ2.

Torque due to weight mg : τ g = −L2mg sinθ1.

τ i=τT+τg=0∑ ⇒ T = mg sinθ1

2sinθ2

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Center of Gravity重心 I

Gravitational force due to a single mass element produces a torque about O: 在重力場裡, 物體上的每一個部分都受到重力. 而這些重力, 相對於某一個支點o, 都會產生力矩. 因此有一合力矩.

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Center of Gravity重心 II •  The gravitational forces acting on all parts of

an object exert a torque on the object.作用在物體上的重力產生的合力矩相當於 –  These forces act like a single force, equal

to the object’s weight, acting at a point called the center of gravity.物體的總重量作用在某一點的力矩, 此點為重心.

–  In a uniform gravitational field, the center of gravity coincides with the center of mass.在均勻重力場裡, 重心就是質心.

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Center of Gravity重心 III

Finding the center of

gravity 如何找到重心（或質心)? 懸吊物體兩次,找到兩個垂直線, 兩線的交點即為重心的位置

The dancer is in static equilibrium. Which point is her center of

gravity?舞者的重心在哪裡? 就是在質心!

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Problem-Solving Strategy: Static Equilibrium

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Example: A Leaning Ladder例題：斜放的梯子 •  At what angle will the ladder slip? 斜放會倒的角度？

The ladder and the forces on it Forces in both directions sum to zero:

The torques are all perpendicular to the plane of the page, so there’s only one torque equation:

Solve the three equations to get

Force, x: fs − n2 = 0, fs ≤ µn1

Force, y: n1 −mg = 0

Torque: Ln2 sinφ − L2mgcosφ = 0

tanφ = sinφcosφ

=mg2 fs

≥12µ

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Stability穩定性Ｉ •  An equilibrium is stable if a slight disturbance from

equilibrium results in forces and/or torques that tend to restore the equilibrium.一平衡狀態被小規模的擾動改變, 所引起的力或力矩會回復其平衡狀態, 此為穩定平衡.

•  An equilibrium is unstable if a slight disturbance causes the system to move away from the original equilibrium.

但是如果引起的力或力矩是使系統遠離原來的平衡狀態, 則為不穩定平衡

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Stability穩定性ＩＩ

Cone on its base is stable Cone on its tip is unstable

•  For example:例如

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Kinds of Stability穩定性的種類

Unstable equilibrium: disturbed ball will leave original equilibrium不穩定平衡

Neutrally stable equilibrium中性穩定平衡(隨處平衡)

Metastable or conditionally stable equilibrium: ball returns for small disturbances, but not for large ones暫穩平衡

Stable equilibrium: disturbed ball will return to equilibrium穩定平衡

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Conditions for Equilibrium and Stability 平衡與穩定的條件I

•  To be in equilibrium, there must be zero net force on an object. –  Therefore the object must be at a maximum or minimum of its

potential energy curve:由上可歸納: 平衡狀態時合力為零,位能是最大值或最小值

•  For stable equilibrium, the object

must be at a minimum:穩定平衡時, 位能為最小值

dUdx

= 0 (condition for equilibrium)

d 2Udx2 > 0 (stable equilibrium)

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Conditions for Equilibrium and Stability 平衡與穩定的條件II

•  The condition for unstable equilibrium is 不穩定平衡是最大值

•  In two and three dimensions, an object can be stable in one direction but not another.

2維,3維時,不同方向的穩定性 也可能不同

2

2 0 (unstable equilibrium)d Udx

<

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Summary •  Static equilibrium requires zero net force and zero net

torque on a system: Fi∑ =

0,

τ i∑ =

ri×Fi∑ =

0

Example: A crane in static equilibrium

•  Equilibria can be stable, unstable, neutrally stable, or metastable.

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Homework

•  11,19, 25, 31 ,55

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=1/6 L

• Problem 54

Chapter 12 Lecture

Slide 12-19

Essential University Physics Richard Wolfson

2nd Edition

© 2012 Pearson Education, Inc.

Static Equilibrium問題集

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•  The figure shows three pairs of forces acting on an object. Which pair, acting as the only forces on the object, results in static equilibrium?

Clicker Question 1

A.  Force pair B.  Force pair C.  Force pair

A1+ A2

B1+ B2

C1+ C2

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•  The dancer in the figure is balanced; that is, she is in static equilibrium. Which of the three lettered points could be her center of gravity?

A.  Point A B.  Point B C.  Point C

Clicker Question 2

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•  The figure shows a person in static equilibrium leaning against a wall. Which one of the following must be true?

Clicker Question 3

A.  There must be a frictional force at the wall, but not necessarily at the floor.

B.  There must be a frictional force at the floor, but not necessarily at the wall.

C.  There must be frictional forces at both wall and floor.

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•  Which of the labeled points in the figure is in metastable equilibrium?

A.  Point A B.  Point B C.  Point C D.  Point D E.  Point E

Clicker Question 4

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•  Which of the labeled points in the figure are in unstable equilibrium?

A.  Points A and C B.  Points A, B, C and E C.  Point A, C and E D.  Points B and D E.  Point E only

Clicker Question 5

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•  A pencil that is balanced on its tip such that it remains stationary

A.  is in stable equilibrium. B.  is in unstable equilibrium. C.  is in metastable equilibrium. D.  has a net torque acting on it. E.  has a net vertical force acting on it.

Clicker Question 6

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•  The center of gravity of an object

A.  is always inside the object. B.  is always outside the object. C.  may be located inside or outside the object.

Clicker Question 7

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•  An object could be in static equilibrium when

A.  only one force is acting on it. B.  two or more forces are acting on it. C.  only one torque is acting on it.

Clicker Question 8