9.2 The second condition for equilibrium  (Page 3/8)

 Page 3 / 8
$\text{net}\phantom{\rule{0.25em}{0ex}}\tau =0$

where net means total. Torques, which are in opposite directions are assigned opposite signs. A common convention is to call counterclockwise (ccw) torques positive and clockwise (cw) torques negative.

When two children balance a seesaw as shown in [link] , they satisfy the two conditions for equilibrium. Most people have perfect intuition about seesaws, knowing that the lighter child must sit farther from the pivot and that a heavier child can keep a lighter one off the ground indefinitely.

She saw torques on a seesaw

The two children shown in [link] are balanced on a seesaw of negligible mass. (This assumption is made to keep the example simple—more involved examples will follow.) The first child has a mass of 26.0 kg and sits 1.60 m from the pivot.(a) If the second child has a mass of 32.0 kg, how far is she from the pivot? (b) What is ${F}_{\text{p}}$ , the supporting force exerted by the pivot?

Strategy

Both conditions for equilibrium must be satisfied. In part (a), we are asked for a distance; thus, the second condition (regarding torques) must be used, since the first (regarding only forces) has no distances in it. To apply the second condition for equilibrium, we first identify the system of interest to be the seesaw plus the two children. We take the supporting pivot to be the point about which the torques are calculated. We then identify all external forces acting on the system.

Solution (a)

The three external forces acting on the system are the weights of the two children and the supporting force of the pivot. Let us examine the torque produced by each. Torque is defined to be

$\tau =\text{rF}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta .$

Here $\theta =90º$ , so that $\text{sin}\phantom{\rule{0.25em}{0ex}}\theta =1$ for all three forces. That means ${r}_{\perp }=r$ for all three. The torques exerted by the three forces are first,

${\tau }_{1}={r}_{1}{w}_{1}$

second,

${\tau }_{2}={–r}_{2}{w}_{\text{2}}$

and third,

$\begin{array}{lll}{\tau }_{\text{p}}& =& {r}_{\text{p}}{F}_{\text{p}}\\ & =& 0\cdot {F}_{\text{p}}\\ & =& 0.\end{array}$

Note that a minus sign has been inserted into the second equation because this torque is clockwise and is therefore negative by convention. Since ${F}_{\text{p}}$ acts directly on the pivot point, the distance ${r}_{\text{p}}$ is zero. A force acting on the pivot cannot cause a rotation, just as pushing directly on the hinges of a door will not cause it to rotate. Now, the second condition for equilibrium is that the sum of the torques on both children is zero. Therefore

${\tau }_{2}={–\tau }_{1},$

or

${r}_{2}{w}_{2}={r}_{1}{w}_{1}.$

Weight is mass times the acceleration due to gravity. Entering ${\mathit{mg}}_{}$ for $w$ , we get

${r}_{2}{m}_{2}g={r}_{1}{m}_{1}g.$

Solve this for the unknown ${r}_{2}$ :

${r}_{2}={r}_{1}\frac{{m}_{1}}{{m}_{2}}.$

The quantities on the right side of the equation are known; thus, ${r}_{2}$ is

${r}_{2}=\left(\text{1.60 m}\right)\frac{\text{26.0 kg}}{\text{32.0 kg}}=\text{1.30 m}.$

As expected, the heavier child must sit closer to the pivot (1.30 m versus 1.60 m) to balance the seesaw.

Solution (b)

This part asks for a force ${F}_{\text{p}}$ . The easiest way to find it is to use the first condition for equilibrium, which is

$\text{net}\phantom{\rule{0.25em}{0ex}}\mathbf{\text{F}}=0.$

The forces are all vertical, so that we are dealing with a one-dimensional problem along the vertical axis; hence, the condition can be written as

the definition of photon
8kg of a hot liquid initial T is 90°© is missed with another liquid 3kg at 20° calculate e équilibrium T
8kg of a hot liquid initial T is 90°© is missed with another liquid 3kg at 20° calculate e équilibrium T
Balki
Bright
what are the products when acid and base mixed?
Austin
what work done
work done is the product of force and distance moved in the direction of force
Work done = force (F) * distance (D)
abdulsalam
what is resounance
Abdul
y
Tracy
explain the three laws of isaac Newton with the reference
1st law ; a body will continue to stay at a state of rest or continue to move at a uniform motion on a straight line unless an external force is been acted upon
Austine
3rd law; in every action there is an equal or opposite reaction
Austine
2nd law: F=ma
Austine
what is circut
newtons law of motion
hasiya
First law:In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.
Manan
is the ability to do work
Energy
Nwany
u from
Hejreen
any body online hain
Hejreen
ability to do work is energy
what is energy
energy is ability of the capacity to doing work
shafiu
what is vector
A quantity that has both magnitude and direction
Donaldo
can a body with out mass float in space
mosco
Is the quantity that has both magnitude and direction
Amoah
Yes it can float in space,e.g.polyethene has no mass that's why it can float in space
Amoah
that's my suggestion,any other explanation can be given also,thanks
Amoah
A charge of 1.6*10^-6C is placed in a uniform electric field in a density 2*5^10Nc^-1, what is the magnitude of the electric force exerted on the charge?
what's phenomena
Phenomena is an observable fact or event.
Love
Prove that 1/d+1/v=1/f
What interference
What is a polarized light called?
Moyinoluwa
what is a half life
the time taken for a radioactive element to decay by half of its original mass
ken
mohammed
Half of the total time required by a radioactive nuclear atom to totally disintegrate
Justice
radioactive elements are those with unstable nuclei(ie have protons more than neutrons, or neutrons more than protons
Justice
in other words, the radioactive atom or elements have unequal number of protons to neutrons.
Justice
state the laws of refraction
Fabian
state laws of reflection
Fabian
Why does a bicycle rider bends towards the corner when is turning?
Mac
When do we say that the stone thrown vertically up wards accelerate negatively?
Mac
Give two importance of insulator placed between plates of a capacitor.
Mac
Macho had a shoe with a big sole moving in mudy Road, shanitah had a shoe with a small sole. Give reasons for those two cases.
Mac
when was the name taken from
retardation of a car
Biola
when was the name retardation taken
Biola
did you mean a motion with velocity decreases uniformly by the time? then, the vector acceleration is opposite direction with vector velocity
Sphere
what's velocity
mosco
Velocity is the rate of change of displacement
Divya