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9.2 The second condition for equilibrium  (Page 5/9)

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Section summary

  • The second condition assures those torques are also balanced. Torque is the rotational equivalent of a force in producing a rotation and is defined to be
    τ = rF sin θ size 12{τ= ital "rF""sin"θ} {}

    where τ size 12{τ} {} is torque, r size 12{r} {} is the distance from the pivot point to the point where the force is applied, F size 12{F} {} is the magnitude of the force, and θ size 12{θ} {} is the angle between F size 12{F} {} and the vector directed from the point where the force acts to the pivot point. The perpendicular lever arm r ⊥ size 12{r rSub { size 8{ ortho } } } {} is defined to be

    r ⊥ = r sin θ size 12{r rSub { size 8{ ortho } } =r"sin"θ} {}

    so that

    τ = r ⊥ F . size 12{τ=r rSub { size 8{ ortho } } F} {}
  • The perpendicular lever arm r ⊥ size 12{r rSub { size 8{ ortho } } } {} is the shortest distance from the pivot point to the line along which F size 12{F} {} acts. The SI unit for torque is newton-meter (N·m) . The second condition necessary to achieve equilibrium is that the net external torque on a system must be zero:
    net τ = 0 size 12{"net "τ=0} {}

    By convention, counterclockwise torques are positive, and clockwise torques are negative.

Conceptual questions

What three factors affect the torque created by a force relative to a specific pivot point?

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A wrecking ball is being used to knock down a building. One tall unsupported concrete wall remains standing. If the wrecking ball hits the wall near the top, is the wall more likely to fall over by rotating at its base or by falling straight down? Explain your answer. How is it most likely to fall if it is struck with the same force at its base? Note that this depends on how firmly the wall is attached at its base.

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Mechanics sometimes put a length of pipe over the handle of a wrench when trying to remove a very tight bolt. How does this help? (It is also hazardous since it can break the bolt.)

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Problems&Exercises

(a) When opening a door, you push on it perpendicularly with a force of 55.0 N at a distance of 0.850m from the hinges. What torque are you exerting relative to the hinges? (b) Does it matter if you push at the same height as the hinges?

a) 46.8 N·m

b) It does not matter at what height you push. The torque depends on only the magnitude of the force applied and the perpendicular distance of the force's application from the hinges. (Children don't have a tougher time opening a door because they push lower than adults, they have a tougher time because they don't push far enough from the hinges.)

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When tightening a bolt, you push perpendicularly on a wrench with a force of 165 N at a distance of 0.140 m from the center of the bolt. (a) How much torque are you exerting in newton × meters (relative to the center of the bolt)? (b) Convert this torque to footpounds.

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Two children push on opposite sides of a door during play. Both push horizontally and perpendicular to the door. One child pushes with a force of 17.5 N at a distance of 0.600 m from the hinges, and the second child pushes at a distance of 0.450 m. What force must the second child exert to keep the door from moving? Assume friction is negligible.

23.3 N

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Use the second condition for equilibrium (net τ = 0) to calculate F p in [link] , employing any data given or solved for in part (a) of the example.

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Repeat the seesaw problem in [link] with the center of mass of the seesaw 0.160 m to the left of the pivot (on the side of the lighter child) and assuming a mass of 12.0 kg for the seesaw. The other data given in the example remain unchanged. Explicitly show how you follow the steps in the Problem-Solving Strategy for static equilibrium.

Given:

m 1 = 26.0 kg, m 2 = 32.0 kg, m s = 12.0 kg, r 1 = 1.60 m, r s = 0.160 m, find (a) r 2 , (b) F p

a) Since children are balancing:

net τ cw = – net τ ccw ⇒ w 1 r 1 + m s gr s = w 2 r 2 alignl { stack { size 12{"net "τ rSub { size 8{"cw"} } =" net "τ rSub { size 8{"ccw"} } } {} #drarrow w rSub { size 8{1} } r rSub { size 8{1} } +m rSub { size 8{s} } ital "gr" rSub { size 8{s} } =w rSub { size 8{2} } r rSub { size 8{2} } {} # {}} } {}

So, solving for r 2 size 12{r rSub { size 8{2} } } {} gives:

r 2 = w 1 r 1 + m s gr s w 2 = m 1 gr 1 + m s gr s m 2 g = m 1 r 1 + m s r s m 2 = ( 26.0 kg ) ( 1.60 m ) + ( 12.0 kg ) ( 0.160 m ) 32.0 kg = 1.36 m

b) Since the children are not moving:

net F = 0 = F p – w 1 – w 2 – w s ⇒ F p = w 1 + w 2 + w s alignl { stack { size 12{"net "F=0=F rSub { size 8{p} } - w rSub { size 8{1} } - w rSub { size 8{2} } - w rSub { size 8{s} } } {} #drarrow F rSub { size 8{p} } =w rSub { size 8{1} } +w rSub { size 8{2} } +w rSub { size 8{3} } {} } } {}

So that

F p = ( 26.0 kg + 32.0 kg + 12.0 kg ) ( 9.80 m / s 2 ) = 686 N alignl { stack { size 12{F rSub { size 8{p} } = left ("26" cdot 0" kg "+" 32" cdot "0 kg "+"12" cdot "0 kg" right ) left (9 cdot "8 " {m} slash {s rSup { size 8{2} } } right )} {} #= {underline {"686"" N"}} {} } } {}

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Test prep for ap courses

Which of the following is not an example of an object undergoing a torque?

  1. A car is rounding a bend at a constant speed.
  2. A merry-go-round increases from rest to a constant rotational speed.
  3. A pendulum swings back and forth.
  4. A bowling ball rolls down a bowling alley.

(a)

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Five forces of equal magnitude, labeled A – E , are applied to the object shown below. If the object is anchored at point P , which force will provide the greatest torque?

The diagram shows a gray that is a rounded shape of a shoe. Slightly to the right of center is a black dot labeled P. There are 5 red arrows nearly perpendicular to the curved line representing the edge of the object. Arrow A is pointing straight down toward where the line curves around the top. Arrow B is a horizontal line pointing away from the object as the edge curves slightly in an almost vertical line. Arrow C is pointing directly below the P from the edge down. At this point the object begins to curve up from the horizontal. Arrow D is pointing toward straight up toward the object's edge where the object is curving upward more than point C. Arrow E is pointing toward the left edge at around a 45 degree angle toward the object and is approximately perpendicular to the object again.
Five forces acting on an object.
  1. Force A
  2. Force B
  3. Force C
  4. Force D
  5. Force E
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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
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Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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