rigidbodies-torqueversione

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Experiment VI: Static Equilibrium of Rigid Bodies: Torques

Goals

• Study the relationship between force, lever arm, and torque; study how torques add

• Study the conditions for static equilibrium of a rigid body

• Study the concept of “center of gravity”

Introduction and Background

Torques: The torque τ created by a force F with respect to a certain axis of rotation is

⊥×= R F τ !"#$where ⊥ R is the so"called lever arm, which is the perpendicular shortest$ distance between the axis

of rotation and the line of force% Torques are vectors% Strictly spea&ing, the direction of a torque

should be perpendicular to either into or out of$ the plane defined by the force and its lever arm%'owever, for simplicity, we will define the direction of a torque as either cloc&wise ()$ or

countercloc&wise (()$ depending which way the torque would ma&e the ob*ect rotate about the

axis of rotation chosen% +ither () or (() can be chosen as the “” direction as long as

consistency is maintained% )e will choose () as the “” direction here% The resultant of multipletorques can then be obtained as the algebraic sum of the individual torques%

Static Equilibrium: -n order for a rigid body to stay in equilibrium, the net force and the nettorque on the ob*ect must both be .ero% )e usually only deal with motion of ob*ects in two

dimensions, in which case the conditions for static equilibrium are

/,/ ==

∑∑ y x F F

!"0$/=∑τ !"1$

The second condition should hold for any choice of the axis of rotation%

Center of Gravity: For a rigid body with finite si.e, the force of gravity acts on all parts of the body% 2ut for the purpose of studying the translational motion of the body as a whole or the static

equilibrium of the body, we can assume that the entire weight of the body acts at a single point%

This point is the center of gravity% )hen we draw the force diagram for a rigid body, we can put asingle force of gravity at the center of gravity%

Experimental Setup

Equipment : 3eter

stic&, fulcrum, clamps,

weights, pan balance

Setup: The experimental

setup is shownschematically in Figure

41

Figure !%# 5 6rrangement for balancing torques%



!"#% The apparatus consists of a meter stic& suspended from a fulcrum% )eights may be hung from

the meter stic& at various positions along the stic& by means of special clamps in order to apply

torques%

Experimental Procedure and ata !nal"sis

A. Torques

#% 6d*ust the location of the fulcrum so that the meter stic&, with no weights hanging on it, isin static equilibrium balance$ in a hori.ontal position% The position of the fulcrum is

roughly the center of gravity of the meter stic&% 7ote that this position is not necessarily in

the middle of the stic& 8/ cm mar&$%

0% )ith a clamp, hang a 0//"g mass on the stic& 0/ cm from the fulcrum% 'ang another 0//"g

mass with a clamp on the opposite side of the stic&% Find a position so that the meter stic& is

again in equilibrium in a hori.ontal position%

9ecord the position of the fulcrum, the masses added to each side of the stic& and their positions to the nearest millimeter mar&%

:eep the first 0//"g fixed and repeat the procedure changing the second mass successivelyto values of 8// g, 4// g, 1// g, 08/ g, #8/ g, and #// g% -n each case ad*ust the position of

the second mass to achieve equilibrium and record the data%

Remember that you need to include the mass of the clamps! Weigh them with a balance and include them in the masses you record.

1% (alculate the weight )$ from the mass%(alculate the lever arm $ from the positions of the mass and the fulcrum%

-nput the values of ) and into +xcel% <lot ) versus ) using +xcel=s inear 9egression

Fit%

Since you have &ept the added torque on one side of the stic& fixed, from the second

condition of static equilibrium you should reali.e that the torque produced by any weight onthe other side must be equal in magnitude and opposite in direction% (hec& whether your

results are consistent with this conclusion by analy.ing the plot of ) versus )%

• )hat value of the slope do you obtain from your computer analysis> )hat is the

expected value> )hat is the ?" uncertainty in the slope> -s your slope the same as the

expected value within the uncertainty>

•

'ow does the intercept compare with the fixed torque on the other side of the stic&>

4% 7ow perform a “inear 9egression Fit” and plot a graph for ) versus #?

• )hat should the slope of this plot give you> )hat is its value>

• )hat is the expected value for the intercept> )hat did you obtain>

• From the results can you any conclusion as to how the force required to balance a fixed

torque depend on its lever arm>

44



B. Resultant of Multiple Torques

#% 3aintain the fulcrum at the same position and balance the meter stic& now with three

weights@ <ut a single weight on one side and the other two on the other side% 9ecord the

masses and the positions% (alculate the magnitude of various torques around the fulcrumand find their vector sum%

• Aoes your vector sum have the value you expect>

C. Center of Gravity:

#% So far the fulcrum has been placed only at the center of gravity% 7ow move the fulcrum to a

point between the #/ cm and 08 cm mar&s% 2alance the meter stic& by applying a singlemass between #// to 8// g to the shorter side% 9ecord the mass, its position, and the

fulcrum position% )eigh the stic& without clamp$ and record its mass%

6lthough the meter stic& has two sections on either side of the fulcrum, according to theconcept of center of gravity we can still regard the entire weight acting at a single point,

center of gravity, and it is the torque generated by this entire weight acting at center of gravity that is balanced by the torque from the hanging weight on the other side% Bse the

second condition for equilibrium to calculate the distance from the center of gravity to the

fulcrum, and thus the position of the center of gravity, and compare this point with the

previously determined position of center of gravity%

0% Araw a free"body force diagram for the meter stic& for the last experimental situation and

answer the following@

•

)hat must be the force exerted by the fulcrum on the meter stick > Show your calculation%

• 9emember the second condition for static equilibrium should hold for any choice of axis

of rotation% 7ow ta&e the / cm mar& as the axis of rotation and calculate the net torque

about it% -s your result as expected within experimental error>

#onclusions

2riefly discuss whether you have accomplished the goals listed at the beginning%

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