Mechanical Equilibrium Conceptual Physics UNIT 1: LINEAR MOTION Chapter 2 Mechanical Equilibrium

I. Force (2.1) A. force– is a push or pull 1. A force is needed to change an object’s state of motion 2. State of motion may be one of two things At rest b. Moving uniformly along a straight-line path.

B. Net force 1. Usually more than one force is acting on an object 2. combination of all forces acting on object is called net force.   3. The net force on an object changes its motion

2.1 Force Net Force The net force depends on the magnitudes and directions of the applied forces.

2.1 Force Net Force The net force depends on the magnitudes and directions of the applied forces.

2.1 Force Net Force The net force depends on the magnitudes and directions of the applied forces.

2.1 Force Net Force The net force depends on the magnitudes and directions of the applied forces.

2.1 Force Net Force The net force depends on the magnitudes and directions of the applied forces.

2.1 Force Net Force The net force depends on the magnitudes and directions of the applied forces.

4. Can add or subtract to get resultant net force 5. If forces acting on object equal zero then we say the net force acting on the object = 0 6. Scientific units for force are Newtons (N)  

C. Tension and Weight 1. Tension is a “stretching force”  

2. When you hang an object from a spring scale the there are two forces acting on object. a. Force of gravity pulling down (also called weight)   b. Tension force pulling upward c. Two forces are equal and opposite in direction and add to zero (net force = 0)

2.1 Force Tension and Weight The upward tension in the string has the same magnitude as the weight of the bag, so the net force on the bag is zero. The bag of sugar is attracted to Earth with a gravitational force of 2 pounds or 9 newtons.

2.1 Force Tension and Weight The upward tension in the string has the same magnitude as the weight of the bag, so the net force on the bag is zero. The bag of sugar is attracted to Earth with a gravitational force of 2 pounds or 9 newtons.

D. Force Vectors 1. Forces can be represented by arrows a. length of arrow represents amount (magnitude) of force b. Direction of arrow represents direction of force c. Refer to arrow as a vector (represents both magnitude and direction of force

2. Vector quantity- needs both magnitude and direction to complete description (i.e. force, velocity, momentum)

3. Scalar quantity- can be described by magnitude only and has no direction (i.e. temperature, speed, distance)

Scaler or Vector quantity? Time interval? scaler Mass? Speed? Velocity? vector Temperature? Displacement? Acceleration?

Arrow-tipped line segment Length represents magnitude Arrow points in specified direction of vector

Vectors are equal if: magnitude and directions are the same Vectors are not equal if: have different magnitude or direction or

Are these vectors equal or not? YES YES NO NO

II. Mechanical Equilibrium (2.2) A. Mechanical equilibrium- a state wherein no physical changes occur (state of steadiness) 1. When net force equals zero, object is said to be in mechanical equilibrium a. Known as equilibrium rule

b. Can express rule mathematically as   1). ∑ symbol stands for “the sum of” 2). F stands for “forces” 2 N -2 N

2.2 Mechanical Equilibrium The sum of the upward vectors equals the sum of the downward vectors. F = 0, and the scaffold is in equilibrium.

2.2 Mechanical Equilibrium The sum of the upward vectors equals the sum of the downward vectors. F = 0, and the scaffold is in equilibrium.

2.2 Mechanical Equilibrium The sum of the upward vectors equals the sum of the downward vectors. F = 0, and the scaffold is in equilibrium.

2.2 Mechanical Equilibrium The sum of the upward vectors equals the sum of the downward vectors. F = 0, and the scaffold is in equilibrium.

2.2 Mechanical Equilibrium think! If the gymnast hangs with her weight evenly divided between the two rings, how would scale readings in both supporting ropes compare with her weight? Suppose she hangs with slightly more of her weight supported by the left ring. How would a scale on the right read?

2.2 Mechanical Equilibrium think! If the gymnast hangs with her weight evenly divided between the two rings, how would scale readings in both supporting ropes compare with her weight? Suppose she hangs with slightly more of her weight supported by the left ring. How would a scale on the right read? Answer: In the first case, the reading on each scale will be half her weight. In the second case, when more of her weight is supported by the left ring, the reading on the right reduces to less than half her weight. The sum of the scale readings always equals her weight.

III. Support Force (2.3) A. support force- the upward force that balances the weight of an object on a surface 1. The upward force balances the weight of an object 2. Support force often called normal force Support force weight

Directions of support force provided by ten bra components: (A) rigid gore, (B) elastic cup, (C) semi-elastic neckline, (D) semi-elastic underarm, (E) rigid wire, (F) rigid cradle, (G) elastic wing, (H) elastic shoulder strap, (I) elastic underband and (J) rigid fastener. 

2. Two forces add mathematically to zero B. For an object at rest on a horizontal surface, the support force must equal the objects weight. 1. Upward force is positive (+) and the downward force is negative (-). 2. Two forces add mathematically to zero 2 N -2 N

IV. Equilibrium of Moving Objects (2.4) A. Equilibrium can exist in both objects at rest and objects moving at constant speed in a straight-line path. 1. Equilibrium means state of no change 2. Sum of forces equal zero

B. Objects at rest are said to be in static equilibrium C. Objects moving at constant speed in a straight-line path are said to be in dynamic equilibrium Static equilibrium dynamic equilibrium Static equilibrium dynamic equilibrium

V. Vectors (2.5) A. Parallel vectors 1. Add vectors if in same direction 2. Subtract vectors if in opposite direction 3. The sum of two or more vectors is called the resultant vector. + = + =

2.5 Vectors The tension in the rope is 300 N, equal to Nellie’s weight.

2.5 Vectors The tension in the rope is 300 N, equal to Nellie’s weight. The tension in each rope is now 150 N, half of Nellie’s weight. In each case, F = 0.

B. Parallel vectors– simple to add or subtract

B. Non-parallel vectors 1. Construct a parallelogram to determine resultant vector 2. The diagonal of the parallelogram shows the resultant   a. Perpendicular vectors R

At one point the lunar lander was traveling 15ft/sec forward while traveling down at 4ft/sec. What was the direction they were actually moving?

Combining vectors that are NOT parallel Result of adding two vectors called the resultant Resultant of two perpendicular vectors is the diagonal of the rectangle with the two vectors as sides Resultant vector

Use simple three step technique to find resultant of a pair of vectors that are at right angles to each other.   First– draw two vectors with their tails touching.

Second-draw a parallel projection of each vector with dashed lines to form a rectangle

Third-draw the diagonal from the point Third-draw the diagonal from the point where the two tails are touching resultant

b. Perpendicular vectors with equal sides (special case) 1). For a square the length of diagonal is or 1.414 2). Resultant = 1.414 x one of sides   1 Resultant = 1

c. Parallelogram (not perpendicular) Construct parallelogram Construct with two vectors as sides Resultant is the diagonal resultant

C. Applying the Parallelogram Rule- as angle increases,  C. Applying the Parallelogram Rule- as angle increases, tension increases.

2.5 Vectors think! Two sets of swings are shown at right. If the children on the swings are of equal weights, the ropes of which swing are more likely to break?

2.5 Vectors think! Two sets of swings are shown at right. If the children on the swings are of equal weights, the ropes of which swing are more likely to break? Answer: The tension is greater in the ropes hanging at an angle. The angled ropes are more likely to break than the vertical ropes.

Pythagorean Theorem- can be used if vectors added are at right angles 90° B

SOHCAHTOA sin, cos, and tan are functions of the angle of a triangle compared to the lengths of the sides of a triangle If you know the distances of the triangle sides, you can determine the inside angles. If you know the angle and one side, you can calculate the length of the other side of the triangle. Remember the following acronym: SOHCAHTOA These will give you the formula depending on which side or angle you need to calculate

Components calculated using following formulas When angles larger than 90°, sign of one or more components may be negative

Remember what the sides of a triangle are called Hypotenuse Opposite θ Adjacent

Example: Length of Opposite side sin 25° = Length of Hypotenuse Length of Opposite side sin 25° = 100 meters length of Opposite side = 38.3 meters 100 meters Opposite = ? θ Adjacent

Graphical Addition of Vectors Simple method for combining vectors to get resultant vector Use ruler to measure length of vector Use protractor to measure angle 55°