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**Equilibrium An object is in “Equilibrium” when:**

There is no net force acting on the object There is no net Torque (we’ll get to this later) In other words, the object is NOT experiencing linear acceleration or rotational acceleration. We’ll get to this later

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**An object is in “Static Equilibrium” when it is**

NOT MOVING.

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**Dynamic Equilibrium An object is in “Dynamic Equilibrium” when it is**

MOVING with constant linear velocity and/or rotating with constant angular velocity.

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**Equilibrium Let’s focus on condition 1: net force = 0**

The x components of force cancel The y components of force cancel

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**Condition 1: No net Force**

We have already looked at situations where the net force = zero. Determine the magnitude of the forces acting on each of the 2 kg masses at rest below. 60° 30°

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**Condition 1: No net Force**

∑Fx = 0 and ∑Fy = 0 mg = 20 N N = 20 N ∑Fy = 0 N - mg = 0 N = mg = 20 N

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**Condition 1: No net Force**

∑Fx = 0 and ∑Fy = 0 T1 = 10 N 20 N T2 = 10 N ∑Fy = 0 T1 + T2 - mg = 0 T1 = T2 = T T + T = mg 2T = 20 N T = 10 N

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**Condition 1: No net Force**

N = mgcos 60 N = 10 N 60° mg =20 N f = 17.4 N N = 10 N Fx - f = 0 f = Fx = mgsin 60 f = 17.4 N

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**Condition 1: No net Force**

∑Fx = 0 and ∑Fy = 0 30° mg = 20 N T1 = 20 N T2 = 20 N T2 = 20 N T2x T2y = 10 N 30° ∑Fx = 0 T2x - T1x = 0 T1x = T2x Equal angles ==> T1 = T2 ∑Fy = 0 T1y + T2y - mg = 0 2Ty = mg = 20 N Ty = mg/2 = 10 N Ty/T = sin 30 T = Ty/sin 30 T = (10 N)/sin30 T = 20 N Note: unequal angles ==> T1 ≠ T2

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**Condition 1: No net Force**

∑Fx = 0 and ∑Fy = 0 30° mg = 20 N T1 = 20 N T2 = 20 N Note: The y-components cancel, so T1y and T2y both equal 10 N

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**Condition 1: No net Force**

∑Fx = 0 and ∑Fy = 0 30° 20 N T1 = 40 N T2 = 35 N Note: The x-components cancel The y-components cancel

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**Condition 1: No net Force A Harder Problem!**

a. Which string has the greater tension? b. What is the tension in each string? 30° 60°

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**a. Which string has the greater tension?**

∑Fx = 0 so T1x = T2x 30° 60° T1 T2 T1 must be greater in order to have the same x-component as T2.

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**What is the tension in each string?**

30° 60° T1 T2 ∑Fx = 0 T2x-T1x = 0 T1x = T2x T1cos60 = T2cos30 ∑Fy = 0 T1y + T2y - mg = 0 T1sin60 + T2sin30 - mg = 0 T1sin60 + T2sin30 = 20 N Solve simultaneous equations! Note: unequal angles ==> T1 ≠ T2

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**Condition 2: net torque = 0**

Torque that makes a wheel want to rotate clockwise is - Torque that makes a wheel want to rotate counterclockwise is + Negative Torque Positive Torque

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**Condition 2: No net Torque**

Weights are attached to 8 meter long levers at rest. Determine the unknown weights below 20 N ??

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**Condition 2: No net Torque**

Weights are attached to an 8 meter long lever at rest. Determine the unknown weight below 20 N ??

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**Condition 2: No net Torque**

Upward force from the fulcrum produces no torque (since r = 0) ∑T’s = 0 T2 - T1 = 0 T2 = T1 F2r2sinø2 = F1r1sinø1 (F2)(4)(sin90) = (20)(4)(sin90) F2 = 20 N … same as F1 r1 = 4 m r2 = 4 m F1 = 20 N F2 =??

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**Condition 2: No net Torque**

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**Condition 2: No net Torque**

Weights are attached to an 8 meter long lever at rest. Determine the unknown weight below 20 N ??

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**Condition 2: No net Torque**

F2 =?? F1 = 20 N r1 = 4 m r2 = 2 m ∑T’s = 0 T2 - T1 = 0 T2 = T1 F2r2sinø2 = F1r1sinø1 (F2)(2)(sin90) = (20)(4)(sin90) F2 = 40 N (force at the fulcrum is not shown)

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**Condition 2: No net Torque**

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**Condition 2: No net Torque**

Weights are attached to an 8 meter long lever at rest. Determine the unknown weight below 20 N ??

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**Condition 2: No net Torque**

F2 =?? F1 = 20 N r1 = 3 m r2 = 2 m ∑T’s = 0 T2 - T1 = 0 T2 = T1 F2r2sinø2 = F1r1sinø1 (F2)(2)(sin90) = (20)(3)(sin90) F2 = 30 N (force at the fulcrum is not shown)

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**Condition 2: No net Torque**

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**In this special case where **

- the pivot point is in the middle of the lever, and ø1 = ø2 F1R1sinø1 = F2R2sinø2 F1R1= F2R2 20 N 40 N 30 N (20)(4) = (20)(4) (20)(4) = (40)(2) (20)(3) = (30)(2)

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**More interesting problems (the pivot is not at the center of mass)**

Masses are attached to an 8 meter long lever at rest. The lever has a mass of 10 kg. Determine the unknown weight below. 20 N ?? CM

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**More interesting problems (the pivot is not at the center of mass)**

Trick: gravity applies a torque “equivalent to” (the weight of the lever)(Rcm) Tcm =(mg)(rcm) = (100 N)(2 m) = 200 Nm 20 N ?? CM Weight of lever Masses are attached to an 8 meter long lever at rest. The lever has a mass of 10 kg.

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**Masses are attached to an 8 meter long lever at rest. **

The lever has a mass of 10 kg. Determine the unknown weight below. F1 = 20 N CM R1 = 6 m R2 = 2 m F2 = ?? Rcm = 2 m Fcm = 100 N ∑T’s = 0 T2 - T1 - Tcm = 0 T2 = T1 + Tcm F2r2sinø2 = F1r1sinø1 + FcmRcmsinøcm (F2)(2)(sin90)=(20)(6)(sin90)+(100)(2)(sin90) F2 = 160 N (force at the fulcrum is not shown)

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Rotational Motion and Equilibrium

Rotational Motion and Equilibrium

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