Unit 4 Statics Static Equilibrium Translational Forces Torque 1
Dynamics vs Statics Dynamics: is the study of forces and motion. We study why objects move. Statics: is the study of forces and NO motion. We study why objects DO NOT move. 2
Recall Newton s First Law All objects remain at rest, or continue to move at a constant velocity unless acted upon by an external unbalanced force. Thus, an object will stay at rest if all of the forces acting on the object balance each other. We say that the object is in a state of STATIC EQUILIBRIUM if it is at rest. 3
An object moving with a constant velocity is in a state of DYNAMIC EQUILIBRIUM. 4
In Static Equilibrium We must consider: Translation Forces Forces that make objects move from one place to another. Rotational Forces Forces that make an object rotate about a point. 5
Translation Forces Activity Draw a free body diagram. What is the mass of the object? 6
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Find the tensions in the ropes 2 sin26o below. T 2 T 1 T 2 cos26o 60.0 kg T 2 T 1 cos42o T 1 Horizontal Forces T Fx 0 1x T2 x 0 T T 1x 2x Vertical Forces Fy T T F o o o o T1 sin 42 T2 sin26 T1 cos 42 T2 cos26 mg o T2 sin26 T1 o sin 42 T o 1 sin42o T2 sin26 o o cos 42 T2 cos26 mg o sin 42 0 1y 2y g 0 T T F 1y 2y g o sin26 o T2 cos 42 cos26 o sin 42 o mg 8
T T 2 2 mg o sin26 o cos 42 cos26 o sin 42 60kg 9.8N kg o sin26 o cos 42 cos26 o sin 42 T2 424N o o T2 sin26 T1 o sin 42 o 424N sin26 sin 42 o o T1 278N 9
OR T 2 T 1 Why do the vectors connect together? The net force is zero 60.0 kg mg T 2 We can find the tensions using the Law of Sines sina sinb sinc a b c T 1 10
Find T 1 We need to find the missing angle first. T 2 T 1 60.0 kg mg T 2 T 1 180 42 26 o o o sin26 T 1 sin112 mg 112 o T1 sin112 mg sin26 T 1 60.0kg 9.8 sin26 N kg sin112 T1 278 N 11
Find T 2 T 2 T 1 sin 42 T 2 sin112 mg 60.0 kg T2 sin112 mg sin 42 mg T 2 T 2 60.0kg 9.8 sin 42 N kg sin112 T2 424 N T 1 12
Find the mass of the snowflake 30.0 o 10.0 N T 1 45.0 o We need to find the missing angle first. o o o 180 45 30 105 o sin 45 sin105 10.0N mg mg 10.0 N mg sin 45 10.0N sin105 m 10.0N sin105 9.8 sin 45 N kg T 1 m 1.39kg 13
Tension Worksheet 14
Torques (Rotational Forces) Archimedes said "Give me a lever long enough and a fulcrum on which to place it, and I shall move the world." 15
Watch what happens when you don t study Torque 16
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Boss man, you ain't gonna believe this! I left my hook hanging in the water while we were at lunch and a BIG fish almost pulled my rig into the water. 25
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Centre of Mass (Gravity Spot) The centre of mass is a single point in a body at which its entire mass is considered to be concentrated. An object can balance on a point only if its center of mass is directly above the point. Alternatively, if you hang an object from a string, the object's center of mass will be directly below the string. 30
It is usually located near the more massive part of an object. The centre of mass is also called the centre of gravity, which is the point at which the force of gravity acts. The force of gravity is equal on both sides of the object s centre of gravity. 31
The center of mass is an important point on an aircraft, as it defines the amount of mass forward or behind the center of gravity that needs to be moved in order to pitch the plane up or down without applying any external forces 32
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Location of Centre of Mass For uniformly, regular shaped objects the centre of mass is at the geometric centre. For example the centre of mass of a ball (sphere) is located at the centre of the ball. For non-uniform shaped objects, the centre of mass is located at its balance point in a gravitational field. 36
Example: Find the centre of mass of the following: 1. A uniform 10 m log? 2. A uniform 15 m extension ladder? 3. A broom? 37
4. A Triangle 38
6.A boomerang? Note: In the case of a boomerang the centre of mass is not located within the actual boomerang! 39
A male versus A female http://hypertextbook.com/facts/2006/centerofmass.shtml 40
A person's center of mass is slightly below his/her belly button, which is nearly the geometric center of a person. Males and females have different centers of mass Females' centers of mass are lower than those of males. The average ratio of center of mass to height in females is approximately 0.543 and the average ratio of center of mass to height in males is approximately 0.560. 41
Where is the centre of mass of a hammer? When a force is not directed through an object s centre of mass, the force will cause the object to rotate. Example: An unbalanced hammer. 42
Look in text pages 233 43
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Torque Torque: The rotational force caused by a force acting at a distance from a pivot point. Examples: Opening a pop bottle Tightening a screw Using brakes in a car Opening a door 45
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Equation for Torque F r where is Torque r is the distance a Force is applied from a pivot point (aka point of rotation) F is the component of the force that acts perpendicular to the surface. What are the units for Torque? Nm 47
Is Torque a vector quantity? Yes! The direction is: clockwise or counterclockwise Clockwise is taken to be positive in the text. Counterclockwise is taken to be positive in mathematics and in engineering. 48
Examples of Torque 1.Find the torque supplied to a 75 cm wide door by the following forces applied at the edge of the door: A) 20N B) 30 N at 30 o to the perpendicular to the door 49
2.Suppose a bolt requires 65 Nm to be properly tightened. What force is required if the wrench is 15 cm long? 50
Not tested on public, but useful info in real life!! 3.In the old British system (and in the US) torque is measured in ft-lbs. How many Nm is one ft-lb? One foot is 0.305 m. One kilogram is 2.2 pounds 51
Static Equilbrium Consists of 2 Parts First Part Translation Forces F F F F F net 1 2 3... F where net is the sum of ALL of the external forces acting on an object through its centre of mass n n Fi 0 i 1 52
Static Equilbrium Consists of 2 Parts Second Part Torques (or Moments of Force) n net 1 2 3... n i 0 i 1 where net is the sum of ALL of the torques (or moments of force) acting on an object. NOTE: These forces are NOT acting through the object s centre of mass. WHY NOT? There is NO rotation force (torque) if a force acts through an object s centre of mass 53
Static Problems Torque (we did translational forces in unit 2) Seesaw Problems Cantilever Problems Crane Problems (or Strut and cable ) Ladder Problems 54
Torque Questions Seesaw problems: 1. Pat and Tyler are playing on a 4 m long seesaw that is supported at the centre. If Pat has a mass of 30 kg and sits at one end of the seesaw, where should Tyler (mass = 35 kg) sit so that the seesaw balances? 55
2.A playground seesaw with a total length of 5.0 m and a mass of 30.0 kg is pivoted at its center. A 20.0 kg child sits on one end of the seesaw. a) Where should a person push with a force of 220 N in order to hold the seesaw level? 56
b) Where should a 40.0 kg child sit to balance the seesaw? 57
3. Consider the diagram below :A load weighing 60 N is to be supported by a force F applied at the end of a 5.0 kg, uniform, lever as shown. What Force is necessary if the fulcrum is placed at position A? 1.2m F A 40cm 60 N 58
4. What force is needed to balance the 10.0 kg mass if: 2.0 m 10.0 kg 1.0 m F A) the seesaw is massless. B) the seesaw has a mass of 42 kg 59
5. In order to hang a mass of M 1 =30.0kg from the horizontal flat roof of a building, a plank of length 2.4 m is placed on the roof. A rock of mass M 2 = 15.0 kg is placed on one end. How far can the end of the plank reach over the building without tipping over? 60
Cantilever Problems 1. A 1.5 x 10 3 kg car is crossing a 120 m long flat bridge which is supported at both ends. When the car is 32 m from one end, what force must each end support be able to provide? A B 61
2. A 4.0 m diving board is supported by two blocks, one at the end and the second 1.0 m from the end. A 60.0 kg person stands at the end of the board. Find the forces given to the board by the two supports if: A) The board is a Canadian Tire special - no mass. A B 62
2. A 4.0 m diving board is supported by two blocks, one at the end and the second 1.0 m from the end. A 60.0 kg person stands at the end of the board. Find the forces given to the board by the two supports if: B) The mass of the board is 40.0 kg. A B 63
3. A uniform rod of mass 13 kg and length 3.0m rests on two points, one at its left end and one at the centre point. What are the contact forces on the rod at these points? Comment on the stability of this situation. The rod is stable provided no mass is added on the unsupported half of the rod 64
4. Two people of unequal strength must carry a uniform beam of length L while holding it horizontal. The weaker of the two holds the beam at one end. A) How far from the other end must the stronger person hold the beam in order to support three quarters of the weight? A B 65
B) Is there a way in which the stronger person can carry the beam at one end and still support more than one half the weight of the beam? 66
5.In the diagram below, a 10.0 m uniform horizontal beam, weighing 1.00 10 2 N is supported by a rope at each end. If a 4.00 10 2 N box is positioned 2.0 m from the left end of the beam, what is the tension in each of the support ropes (T 1 andt 2 )? (June 2005 - Public) 67
Crane Problems Popular Public Questions (or Suspended load from a strut and cable) 1. Determine the tension in the cable and the compression force in the boom to support the 1.0 x 10 2 kg object. The angle between the boom and the supporting cable is 37. 68
2. A crane is used to lift a uniform 24 m long pipe with a mass of 730 kg as shown in the diagram below. What minimum tension is required by the cable to lift the end of the pipe off the ground? 69
3.A traffic light hangs from a structure as shown. The uniform aluminum pole AD is 4.0 m long and weighs 5.0kg. The weight of the traffic light is 10.0 kg. A)Determine the tension in the horizontal, massless cable CD. B) Determine the vertical and horizontal components of the force exerted by the pivot A on the aluminum pole 70
4. A uniform rod of length L and mass of 4.0 kg is hinged at the left end. A 25.0 kg sign is suspended from the right end. A guy wire is connected to the end of the rod and fastened to a wall. A) Draw a free body diagram for the rod. B) Determine the tension in the guy wire. 71
June 2004 Public (i) Sketch the free body diagram for the rod in the diagram above. Label all forces. (2 marks) (ii) If the mass of the block is 5.0 kg and the rod is uniform with a mass of 0.40 kg, what is the magnitude of the tension in the wire? (3 marks) Answer 72
Ladder Problems For problems involving a ladder (or other object) leaning against a wall, the wall should be considered frictionless. These problems should be limited to three forces acting in different directions (i.e., three different sets of components). 73
1. A 8.2 kg ladder is resting against a wall such that the angle made with the ground is 75 o. Find the force friction required by the ground to keep the ladder from moving. 74
2. A 4.0 m ladder is resting against a building such that the foot of the ladder is 0.75 m away from the building. The ladder weighs 99 N. Find the force of friction required at ground level to keep the ladder from sliding if a 75 kg person stands at following locations along the ladder: A) 1.0 m from the bottom of the ladder. 75
2. A 4.0 m ladder is resting against a building such that the foot of the ladder is 0.75 m away from the building. The ladder weighs 99 N. Find the force of friction required at ground level to keep the ladder from sliding if a 75 kg person stands at following locations along the ladder: B) at the middle of the ladder. 76
2. A 4.0 m ladder is resting against a building such that the foot of the ladder is 0.75 m away from the building. The ladder weighs 99 N. Find the force of friction required at ground level to keep the ladder from sliding if a 75 kg person stands at following locations along the ladder: C) 3.0 m from the bottom. 77
2. A 4.0 m ladder is resting against a building such that the foot of the ladder is 0.75 m away from the building. The ladder weighs 99 N. Find the force of friction required at ground level to keep the ladder from sliding if a 75 kg person stands at following locations along the ladder: D) 3.7 m from the bottom 78
3. A 5.0 m long ladder leans against a wall at a point 4.0 m above the ground as shown. The ladder is uniform and has a mass of 12.0 kg. A 55 kg painter is standing 3.0 m up the ladder. Assuming the wall is frictionless (but the ground is not) determine the forces exerted on the ladder by the ground and the wall. 79
http://surendranath.tripod.com/applets/ Dynamics/CM/CMApplet.html 80