Torque and Angular Momentum
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Torque is a Torque is a twist or turn twist or turn that tends to that tends to produce produce rotation. * * * rotation. * * * Applications are Applications are found in many found in many common tools common tools around the home around the home or industry or industry where it is where it is necessary to necessary to turn, tighten or turn, tighten or loosen devices. loosen devices. Understanding Torque
Transcript of Torque and Angular Momentum
Torque is a twist or Torque is a twist or turn that tends to turn that tends to produce rotation. * produce rotation. * * * * * Applications are Applications are found in many found in many common tools common tools around the home around the home or industry where it or industry where it is necessary to is necessary to turn, tighten or turn, tighten or loosen devices.loosen devices.
Understanding Torque
Definition of TorqueDefinition of Torque
Torque is defined as the tendency to produce a change in rotational motion.
Torque is defined as the tendency to produce a change in rotational motion.
Examples:
Torque is Determined by Three Factors:
Torque is Determined by Three Factors:
l The The magnitudemagnitude of the applied force. of the applied force.l The The directiondirection of the applied force. of the applied force.l The The locationlocation of the applied force. of the applied force.
l The The magnitudemagnitude of the applied force. of the applied force.l The The directiondirection of the applied force. of the applied force.l The The locationlocation of the applied force. of the applied force.
20 N
Magnitude of force
40 N
The 40-N force produces twice the torque as does the
20-N force.
Each of the 20-N forces has a different
torque due to the direction of force. 20 N
Direction of Force
20 N
20 N20 N
Location of forceThe forces nearer the
end of the wrench have greater torques.
20 N20 N
Units for TorqueUnits for TorqueTorque is proportional to the magnitude of F and to the distance r from the axis. Thus, a tentative formula might be:
Torque is proportional to the magnitude of F and to the distance r from the axis. Thus, a tentative formula might be:
= Fr = Fr Units: Nm or lbft
6 cm
40 N
= (40 N)(0.60 m)
= 24.0 Nm, cw
= 24.0 Nm, cw = 24.0 Nm, cw
Direction of TorqueDirection of Torque
Torque is a vector quantity that has direction as well as magnitude.
Torque is a vector quantity that has direction as well as magnitude.
Turning the handle of a screwdriver clockwise
and then counterclockwise will
advance the screw first inward and then outward.
Sign Convention for TorqueSign Convention for TorqueBy convention, counterclockwise
torques are positive and clockwise torques are negative.
Positive torque: Counter-
clockwise, out of page
cw
ccw
Negative torque: clockwise, into page
Line of Action of a ForceLine of Action of a Force
The line of action of a force is an imaginary line of indefinite length drawn along the direction of the force.
The line of action of a force is an imaginary line of indefinite length drawn along the direction of the force.
F1
F2
F3Line of action
The Moment ArmThe Moment Arm
The moment arm of a force is the perpendicular distance from the line of action of a force to the axis of rotation.
The moment arm of a force is the perpendicular distance from the line of action of a force to the axis of rotation.
F2
F1
F3
r
rr
Calculating TorqueCalculating Torquel Read problem and draw a rough Read problem and draw a rough
figure.figure.l Extend line of action of the force.Extend line of action of the force.l Draw and label moment arm.Draw and label moment arm.l Calculate the moment arm if Calculate the moment arm if
necessary.necessary.l Apply definition of torque:Apply definition of torque:
l Read problem and draw a rough Read problem and draw a rough figure.figure.
l Extend line of action of the force.Extend line of action of the force.l Draw and label moment arm.Draw and label moment arm.l Calculate the moment arm if Calculate the moment arm if
necessary.necessary.l Apply definition of torque:Apply definition of torque:
= Fr = Fr Torque = force x moment arm
Example 1: An 80-N force acts at the end of a 12-cm wrench as shown. Find the torque.
Example 1: An 80-N force acts at the end of a 12-cm wrench as shown. Find the torque.
• Extend line of action, draw, calculate r.
= (80 N)(0.104 m) = 8.31 N m
= (80 N)(0.104 m) = 8.31 N m
r = 12 cm sin 600 = 10.4 cm
r = 12 cm sin 600 = 10.4 cm
Alternate: An 80-N force acts at the end of a 12-cm wrench as shown. Find the torque.
Alternate: An 80-N force acts at the end of a 12-cm wrench as shown. Find the torque.
Resolve 80-N force into components as shown.
Note from figure: rx = 0 and ry = 12 cm
= (69.3 N)(0.12 m) = 8.31 N m as before = 8.31 N m as before
positive
12 cm
Calculating Resultant TorqueCalculating Resultant Torque• Read, draw, and label a rough figure.Read, draw, and label a rough figure.
• Draw free-body diagram showing all Draw free-body diagram showing all forces, distances, and axis of rotation.forces, distances, and axis of rotation.
• Extend lines of action for each force.Extend lines of action for each force.
• Calculate moment arms if necessary.Calculate moment arms if necessary.
• Calculate torques due to EACH individual Calculate torques due to EACH individual force affixing proper sign. CCW (+) and force affixing proper sign. CCW (+) and CW (-).CW (-).
• Resultant torque is sum of individual Resultant torque is sum of individual torques.torques.
• Read, draw, and label a rough figure.Read, draw, and label a rough figure.
• Draw free-body diagram showing all Draw free-body diagram showing all forces, distances, and axis of rotation.forces, distances, and axis of rotation.
• Extend lines of action for each force.Extend lines of action for each force.
• Calculate moment arms if necessary.Calculate moment arms if necessary.
• Calculate torques due to EACH individual Calculate torques due to EACH individual force affixing proper sign. CCW (+) and force affixing proper sign. CCW (+) and CW (-).CW (-).
• Resultant torque is sum of individual Resultant torque is sum of individual torques.torques.
Example 2: Find resultant torque about axis A for the arrangement shown below:
Example 2: Find resultant torque about axis A for the arrangement shown below:
300300
6 m 2 m4 m
20 N30 N
40 NA
Find due to each force.
Consider 20-N force first:
Find due to each force.
Consider 20-N force first:
r = (4 m) sin 300 = 2.00 m
= Fr = (20 N)(2 m) = 40 N m, cw
The torque about A is clockwise and
negative.
20 = -40 N m20 = -40 N m
r
negative
Example 2 (Cont.): Next we find torque due to 30-N force about same axis A.Example 2 (Cont.): Next we find torque due to 30-N force about same axis A.
300300
6 m 2 m4 m
20 N30 N
40 NA
Find due to each force.
Consider 30-N force next.
Find due to each force.
Consider 30-N force next.
r = (8 m) sin 300 = 4.00 m
= Fr = (30 N)(4 m) = 120 N m, cw
The torque about A is clockwise and
negative.
30 = -120 N m30 = -120 N m
rnegative
Example 2 (Cont.): Finally, we consider the torque due to the 40-N force.Example 2 (Cont.): Finally, we consider the torque due to the 40-N force.
Find due to each force.
Consider 40-N force next:
Find due to each force.
Consider 40-N force next:
r = (2 m) sin 900 = 2.00 m
= Fr = (40 N)(2 m) = 80 N m, ccw
The torque about A is CCW and positive.
40 = +80 N m40 = +80 N m
300300
6 m 2 m4 m
20 N30 N
40 NA
r
positive
Example 2 (Conclusion): Find resultant torque about axis A for the arrangement shown below:
Example 2 (Conclusion): Find resultant torque about axis A for the arrangement shown below:
300300
6 m 2 m4 m
20 N30 N
40 NA
Resultant torque is the sum of individual torques.
Resultant torque is the sum of individual torques.
R = - 80 N mR = - 80 N m Clockwise
R = 20 + 20 + 20 = -40 N m -120 N m + 80 N m
Part II: Torque and the Cross Product or Vector Product.
Part II: Torque and the Cross Product or Vector Product.
Optional Discussion
This concludes the general treatment of torque. Part II details the use of the vector product in calculating resultant torque. Check with your instructor before studying this section.
The Vector ProductThe Vector Product
Torque can also be found by using the vector product of force F and position vector r. For example, consider the figure below.
F
r
F Sin The effect of the
force F at angle (torque) is to advance the bolt out of the page.
Torque
Magnitude:
(F Sin )r
Direction = Out of page (+).
Definition of a Vector Product
Definition of a Vector ProductThe magnitude of the vector (cross)
product of two vectors A and B is defined as follows:
A x B = l A l l B l Sin
F x r = l F l l r l Sin Magnitude onlyMagnitude only
F
(F Sin ) r or F (r Sin )
In our example, the cross product of F and r is:
In effect, this becomes simply:
r
F Sin
Example: Find the magnitude of the cross product of the vectors r and F drawn below:
Example: Find the magnitude of the cross product of the vectors r and F drawn below:
r x F = l r l l F l Sin r x F = (6 in.)(12 lb) Sin
r x F = l r l l F l Sin
r x F = (6 in.)(12 lb) Sin 120
Explain Explain differencedifference.. Also, what aboutAlso, what about F x r??
12 lb
r x F = 62.4 lb in.
Torque
600
6 in.
Torque
600
6 in.
12 lb r x F = 62.4 lb in.
Direction of the Vector Product.Direction of the Vector Product.
The The directiondirection of of a vector product a vector product is determined by is determined by the the right hand right hand rule.rule. A
C
BB
-CA
A x B = C (up)A x B = C (up)
B x A = -C B x A = -C (Down)(Down)
Curl fingers of right Curl fingers of right hand in direction of hand in direction of cross pro-duct (cross pro-duct (AA to to BB) ) or (or (BB to to AA). ). ThumbThumb will will point in the direction of point in the direction of product product CC..
What is What is direction of A x direction of A x
C?C?
Example: What are the magnitude and direction of the cross product, r x F?Example: What are the magnitude and direction of the cross product, r x F?
r x F = l r l l F l Sin r x F = (6 in.)(10 lb) Sin
r x F = 38.3 lb in.
10 lbTorqu
e
500
6 in. Magnitude
Out
r
FDirection by right hand rule:Out of paper (thumb) or +kr x F = (38.3 lb in.) k
What are magnitude and direction of F x r?
SummarySummaryTorque is the product of a force and its
moment arm as defined below:Torque is the product of a force and its
moment arm as defined below:
The moment arm of a force is the perpendicular distance from the line of action of a force to the axis of rotation.
The moment arm of a force is the perpendicular distance from the line of action of a force to the axis of rotation.
The line of action of a force is an imaginary line of indefinite length drawn along the direction of the force.
The line of action of a force is an imaginary line of indefinite length drawn along the direction of the force.
= Fr = Fr Torque = force x moment arm
Torque = force x moment arm
Summary: Resultant TorqueSummary: Resultant Torque• Read, draw, and label a rough figure.Read, draw, and label a rough figure.
• Draw free-body diagram showing all Draw free-body diagram showing all forces, distances, and axis of rotation.forces, distances, and axis of rotation.
• Extend lines of action for each force.Extend lines of action for each force.
• Calculate moment arms if necessary.Calculate moment arms if necessary.
• Calculate torques due to EACH individual Calculate torques due to EACH individual force affixing proper sign. CCW (+) and force affixing proper sign. CCW (+) and CW (-).CW (-).
• Resultant torque is sum of individual Resultant torque is sum of individual torques.torques.
• Read, draw, and label a rough figure.Read, draw, and label a rough figure.
• Draw free-body diagram showing all Draw free-body diagram showing all forces, distances, and axis of rotation.forces, distances, and axis of rotation.
• Extend lines of action for each force.Extend lines of action for each force.
• Calculate moment arms if necessary.Calculate moment arms if necessary.
• Calculate torques due to EACH individual Calculate torques due to EACH individual force affixing proper sign. CCW (+) and force affixing proper sign. CCW (+) and CW (-).CW (-).
• Resultant torque is sum of individual Resultant torque is sum of individual torques.torques.
