# Chapter 5A. Torque. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University

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1 Chapter 5A. Torque A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University 2007

2 Torque is a twist or turn that tends to produce rotation. * * * Applications are found in many common tools around the home or industry where it is necessary to turn, tighten or loosen devices.

3 Objectives: After completing this module, you should be able to: Define and give examples of the terms torque, moment arm, axis, and line of action of a force. Draw, label and calculate the moment arms for a variety of applied forces given an axis of rotation. Calculate the resultant torque about any axis given the magnitude and locations of forces on an extended object. Optional: Define and apply the vector cross product to calculate torque.

4 Definition of Torque Torque is is defined as as the tendency to to produce a change in in rotational motion. Examples:

5 Torque is Determined by Three Factors: The magnitude of the applied force. The direction of the applied force. The location of the applied force. The Each The forces 40-N of the nearer force 20-N the forces produces end of has the a twice wrench different the have torque greater as due does torques. the direction 20-N force. of force. Direction Magnitude Location of of of Force force 20 N 20 N20 NN N N 20 N20 N

6 Units for Torque Torque is is proportional to to the magnitude of of F and to to the distance r from the axis. Thus, a tentative formula might be: = Fr Units: N m or lb ft = (40 N)(0.60 m) = 24.0 N m, cw = 24.0 N m, cw 6 cm 40 N

7 Direction of Torque Torque is is a vector quantity that has direction as as well as as magnitude. Turning the handle of a screwdriver clockwise and then counterclockwise will advance the screw first inward and then outward.

8 Sign Convention for Torque By convention, counterclockwise torques are positive and clockwise torques are negative. Positive torque: Counter-clockwise, out of page cw ccw Negative torque: clockwise, into page

9 Line of Action of a Force The The line of of action of of a force is is an an imaginary line line of of indefinite length drawn along the the direction of of the the force. F 1 F 2 F 3 Line of action

10 The Moment Arm The The moment arm of of a force is is the the perpendicular distance from the the line line of of action of of a force to to the the axis of of rotation. F 1 r F 2 r r F 3

11 Calculating Torque Read problem and draw a rough figure. Extend line of action of the force. Draw and label moment arm. Calculate the moment arm if if necessary. Apply definition of torque: = Fr Torque = force x moment arm

12 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. r = 12 cm sin 60 0 = 10.4 cm = (80 N)(0.104 m) = 8.31 N m

13 Alternate: An 80-N force acts at the end of a 12-cm wrench as shown. Find the torque. positive 12 cm Resolve 80-N force into components as shown. Note from figure: r x = 0 and r y = 12 cm = (69.3 N)(0.12 m) = 8.31 N m as before

14 Calculating Resultant Torque Read, draw, and label a rough figure. Draw free-body diagram showing all forces, distances, and axis of of rotation. Extend lines of of action for each force. Calculate moment arms if if necessary. Calculate torques due to to EACH individual force affixing proper sign. CCW (+) and CW (-).( Resultant torque is is sum of of individual torques.

15 Example 2: Find resultant torque about axis A for the arrangement shown below: Find due to to each force. Consider 20-N force first: r = (4 m) sin 30 0 = 2.00 m = Fr = (20 N)(2 m) = 40 N m, cw 30 N m 2 m 40 N negative r A m 20 N The torque about A is clockwise and negative = -40 N m

16 Example 2 (Cont.): Next we find torque due to 30-N force about same axis A. Find due to to each force. Consider 30-N force next. r = (8 m) sin 30 0 = 4.00 m = Fr = (30 N)(4 m) = 120 N m, cw 30 N m 2 m 40 N r negative A m 20 N The torque about A is clockwise and negative = -120 N m

17 Example 2 (Cont.): Finally, we consider the torque due to the 40-N force. Find due to to each force. Consider 40-N force next: r = (2 m) sin 90 0 = 2.00 m = Fr = (40 N)(2 m) = 80 N m, ccw 30 N m 2 m 40 N positive r A m The torque about A is CCW and positive = +80 N m 20 N

18 Example 2 (Conclusion): Find resultant torque about axis A for the arrangement shown below: Resultant torque is is the sum of of individual torques. 30 N m 2 m 40 N A m 20 N R = = -40 N m -120 N m + 80 N m R = - 80 N m Clockwise

19 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.

20 The 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. Torque r Magnitude: (F Sin )r F Sin F The effect of the force F at angle (torque) is to advance the bolt out of the page. Direction = Out of page (+).

21 Definition of a Vector Product The magnitude of the vector (cross) product of two vectors A and B is defined as follows: AxB= l A l l B l Sin In our example, the cross product of F and r is: F x r = l F l l r l Sin Magnitude only F Sin r F In effect, this becomes simply: (F Sin ) r or F (r Sin )

22 Example: Find the magnitude of the cross product of the vectors r and F drawn below: Torque 12 lb r x F = l r l l F l Sin 6 in. 6 in r x F = (6 in.)(12 lb) Sin r x F = 62.4 lb in. r x F = l r l l F l Sin Torque 60 0 r x F = (6 in.)(12 lb) Sin lb r x F = 62.4 lb in. Explain difference. Also, what about F x r?

23 Direction of the Vector Product. The direction of a vector product is determined by the right hand rule. A x B = C (up) B x A = -C C (Down) What is direction of A x C? C B A A -C Curl fingers of right hand in direction of cross pro- duct (A( to B) ) or (B( to A). Thumb will point in the direction of product C. B

24 Example: What are the magnitude and direction of the cross product, r x F? Torque 10 lb r x F = l r l l F l Sin Out 50 0 r x F = (6 in.)(10 lb) Sin 6 in. r x F = 38.3 lb in. Magnitude F r Direction by right hand rule: Out of paper (thumb) or +k r x F = (38.3 lb in.) k What are magnitude and direction of F x r?

25 Cross Products Using (i,j,k( i,j,k) k y j z i i Magnitudes are zero for parallel vector products. i Consider 3D axes (x, y, z) x Define unit vectors, i, j, k Consider cross product: i x i i x i = (1)(1) Sin 0 0 = 0 j x j = (1)(1) Sin 0 0 = 0 k x k = (1)(1)Sin 0 0 = 0

26 Vector Products Using (i,j,k( i,j,k) z k y j i j i x Magnitudes are 1 for perpendicular vector products. Consider 3D axes (x, y, z) Define unit vectors, i, j, k Consider dot product: i x j i x j = (1)(1) Sin 90 0 = 1 j x k = (1)(1) Sin 90 0 = 1 k x i = (1)(1) Sin 90 0 = 1

27 Vector Product (Directions) k y j i x Directions are given by the right hand rule. Rotating first vector into second. z j i x j = (1)(1) Sin 90 0 = +1 k k i j x k = (1)(1) Sin 90 0 = +1 i k x i = (1)(1) Sin 90 0 = +1 j

28 Vector Products Practice (i,j,k( i,j,k) z k y j j i x Directions are given by the right hand rule. Rotating first vector into second. i x k =? k x j =? -j (down) -i (left) k j x -i =? i 2 i x -3 k =? + k (out) + 6 j (up)

29 Using i,j Notation - Vector Products Consider: A = 2 i - 4 j and B = 3 i + 5 j A x B = (2 i - 4 j) x (3 i + 5 j) = 0 k -k 0 (2)(3) ixi + (2)(5) ixj + (-4)(3) jxi + (-4)(5) jxj A x B = (2)(5) k + (-4)(3)(-k) = +22 k Alternative: A = 2 i - 4 j B = 3 i + 5 j A x B = 10 - (-12) = +22 k Evaluate determinant

30 Summary Torque is is the product of a force and its moment arm as defined below: The The moment arm of of a force is is the the perpendicular distance from the the line line of of action of of a force to to the the axis of of rotation. The The line of of action of of a force is is an an imaginary line line of of indefinite length drawn along the the direction of of the the force. = Fr Torque = force x moment arm

31 Summary: Resultant Torque Read, draw, and label a rough figure. Draw free-body diagram showing all forces, distances, and axis of of rotation. Extend lines of of action for each force. Calculate moment arms if if necessary. Calculate torques due to to EACH individual force affixing proper sign. CCW (+) and CW (-).( Resultant torque is is sum of of individual torques.

32 CONCLUSION: Chapter 5A Torque

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