## Forces in Two-Dimensions - Detailed Help

### Assignment 2D4: Static Equilibrium Analysis

Objectives:
• The student should be able to physically analyze situations in which a weight is supported by two or more cables which extend at angles to the horizontal.
• The student should be able to perform calculations to mathematically relate the tension in a sign-supporting cable to the angle with the horizontal and the mass of the sign.

The Physics Classroom, Vectors and Motion in Two Dimensions Unit, Lesson 3, Part b

 A sign with a mass of m is hung symmetrically from two cables which make an angle of theta with the horizontal (see diagram). The tension in each one of the cables is Ftens. The vertical component of the tension force is Fy; the horizontal component is Fx. Which of the following mathematical statements are true? Definition of Equilibrium: Equilibrium is the condition in which all the individual forces acting upon an object are balanced. If an object is at equilibrium, then the individual forces acting upon the object are balanced. There is no unbalanced force - the net force is 0 Newtons. All the vertical forces balance and all the horizontal forces balance. In this situation, there are two forces which are at angles to the vertical and horizontal (the tension forces). It is the usual strategy to first resolve any force at angles to the axis into components or parts which are directed along the axis. Thus, each tension force has a horizontal component (Fx) and a vertical component (Fy). The two vertical components are directed upward and together they must balance the downward force of gravity. The two horizontal components are directed left and right and must balance each other. The tension force has a horizontal and an upward component or effect on the sign. The horizontal component (Fx) can be calculated as Ftens• cosine() where is the angle which the force makes with the horizontal. The upward component (Fy) can be calculated as Ftens• sine() where is the angle which the force makes with the horizontal. Fx = Ftens• cosine() Fy = Ftens • sine() The force of gravity (Fgrav) acting upon an object can be determined from the mass of an object using the equation: Fgrav = mass • g where g is the acceleration caused by gravity alone. The value of g on Earth is 9.8 m/s/s.

 A sign is hung symmetrically from two cables which make an angle with the horizontal (see diagram). The sign is in a state of static equilibrium. Which of the following statements are true? List all that apply ... Definition of Equilibrium: Equilibrium is the condition in which all the individual forces acting upon an object are balanced. If an object is at equilibrium, then the individual forces acting upon the object are balanced. There is no unbalanced force - the net force is 0 Newtons. All the vertical forces balance and all the horizontal forces balance. When added as vectors, they must add to 0 N. There are two cables directed at angles to the horizontal and vertical axes. These cables exert tension forces on the sign and have horizontal and vertical components. The only other force acting on the sign is the force of gravity. The two vertical components of tension must act together to balance the force of gravity. The horizontal components of the tension force are the only horizontal forces and must balance each other. Some students have the improper habit of thinking that an object at equilibrium is an object upon which all forces are equal. After all, the word equilibrium sounds like the word equal. But don't be fooled! Equilibrium has to do with a balance of forces. While it is possible that all the forces might be equal, they don't have to be equal to be balanced. For instance, an object with an up force of 20 N and a down force of 20 N and a right force of 8 N and a left force of 8 N is at equilibrium. Are all the forces balanced? Yes! Are all the forces equal? No! The 20 N is not equal to the 8 N.

 A sign which weighs 100 N is supported symmetrically by two cables which make an angle of 20.0 degrees with the horizontal. A single cable will pull upward on the sign with a force of ____ Newtons. (Note: Numbers are randomized numbers and likely different from the numbers listed here.) Definition of Equilibrium: Equilibrium is the condition in which all the individual forces acting upon an object are balanced. If a sign is at equilibrium, then the individual forces acting upon the sign are balanced. Thus, there is no unbalanced force - the net force is 0 Newtons. In order for the net force to be 0 N, there must be a balance of both horizontal and vertical forces. If the sign is supported by two cables, each cable must pull with sufficient enough upward force to balance the weight of the sign (the downward force of gravity). Thus, if the sign weighs 100 N, the upward components of tension in each cable must together provide 100 N of upward force.

 A flower pot which weighs 20.0 N is supported symmetrically by three cables which make an angle of 60.0 degrees with the horizontal. A single cable will pull upward on the flower pot with a force of ____ Newtons. (Note: Numbers are randomized numbers and likely different from the numbers listed here.) Definition of Equilibrium: Equilibrium is the condition in which all the individual forces acting upon an object are balanced. If a flower pot is at equilibrium, then the individual forces acting upon the pot are balanced. The net force is 0 Newtons. In order for the net force to be 0 N, there must be a balance of both horizontal and vertical forces. If the flower pot is supported by cables, all the cables together must pull with sufficient enough upward force to balance the weight (the downward force of gravity) of the flower pot. Thus, if the pot weighs 100 N and there are four cables, the upward components of tension in each cable must together provide 100 N of upward force. If the cables are arranged symmetrically, then the weight of the pot is distributed equally among the four cables. Each of the four cables will have a vertical component of 25 N to supply the needed 100 N of upward force to balance the pot's weight. The question asks to determine the amount of upward pull exerted by the cable. The word upward is critical to the question. The question is focused on the vertical component of tension in the sign - not simply on the total amount of tension in the sign.

 A sign is hung by two cables, each of which makes an angle of theta with the horizontal. As the angle theta is INCREASED (or DECREASED), the weight of the sign ____; the tension force in the cable ____; and the vertical component of the tension force ____. Definition of Equilibrium: Equilibrium is the condition in which all the individual forces acting upon an object are balanced. If the sign remains at equilibrium, then all three individual forces which are acting upon it must remain balanced as the angle is changed. A change in the angle will affect the amount of horizontal pull in the cable which in turn affects the amount of tension in the cable. The more horizontally aligned the cable is, the more it will pull horizontally. Thus, a decrease in the angle will increase the horizontal component of tension and an increase in the angle will decrease the horizontal component of tension. These changes in the horizontal components will result in the same change in the overall tension in the cable. Some students who have difficulty with the concept of weight will have difficulty with this question. The weight of a sign depends upon its mass. Don't be fooled! Changes in the cables which support the sign will not effect the sign's weight. Other difficulties with this question relate to the vertical component of the tension force. For a sign hung symmetrically by two cables, the weight of the sign is distributed equally to the two cables. Thus, the upward pull (vertical only) of the cables is one-half the weight of the sign. Changes in the angle will affect the horizontal component of tension; but the vertical component of tension must be of sufficient value to balance one-half the weight of the sign.

 Suppose you are hanging a picture by two cables in your living room. You are considering three different angle orientations as shown. Which orientation - A, B or C - would result in the least tension (or greatest tension) in the cables? Definition of Equilibrium: Equilibrium is the condition in which all the individual forces acting upon an object are balanced. If the picture is at equilibrium, then all three individual forces which are acting upon it must be balanced regardless of the angle. A change in the angle will affect the amount of horizontal pull in the cable which in turn affects the amount of tension in the cable. The more horizontally aligned the cable is, the more it will pull horizontally. This increased horizontal pull will increase the tension in the cable.

 A sign with a mass of 3.66 kg is being hung symmetrically by two cables which make an angle of 37.2 degrees with the horizontal. Draw a free-body diagram and perform a trigonometric analysis to determine the tension in one of the cables. (Note: Your numbers are selected at random and likely different from the numbers listed here.) Success at a problem in physics is dependent upon a carefully plotted strategy. The strategy below will prove useful in this question: Construct a free-body diagram for the sign. Represent each of the three forces by vector arrows which point in the direction of each force; label the forces according to their type. Use the mass to calculate the downward force of gravity (see Formula Fix section below). Determine the vertical component of the tension (Fy) in each cable. See Think About It section below. Sketch a force triangle and label the sides - Ftens for the hypotenuse and Fy for the vertical side. Label the angle . See Math Magic section below. Using a trigonometric function, calculate the tension force from knowledge of the angle and the Fy value. See Math Magic section below. All the individual forces acting upon the sign must balance. The cables are at an angle, so each cable has a vertical and a horizontal component of tension. Since the sign is hung symmetrically, the weight of the sign is distributed equally to each cable. Thus, the vertical component of tension is the same in each cable and equal to one-half the weight of the sign. The tension in the cable is a force vector. Vectors are represented by vector arrows. Vectors such as this one have horizontal and vertical components. The components are often represented by constructing a right triangle about the vector such that the vector is the hypotenuse of the right triangle. The components are then the legs of the right triangle. Trigonometric functions can be used to relate the values of the components to the value of the vector. The legendary SOH CAH TOA is applied to the force triangle in this question to give the following results. Fx = Ftens • cosine Fy = Ftens • sine where = angle between the cable and the horizontal The force of gravity (Fgrav) acting upon an object can be determined from the mass of an object using the equation: Fgrav = mass • g where g is the acceleration caused by gravity alone. The value of g on Earth is 9.8 m/s/s.

 A sign which weighs 42.6 N is being hung symmetrically by two cables which make an angle of 17.5 degrees with the horizontal. Draw a free-body diagram and perform a trigonometric analysis to determine the tension in one of the cables. (Note: Your numbers are selected at random and likely different from the numbers listed here.) Success at a problem in physics is dependent upon a carefully plotted strategy. The strategy below will prove useful in this question: Construct a free-body diagram for the sign. Represent each of the three forces by vector arrows which point in the direction of each force; label the forces according to their type. Determine the vertical component of the tension (Fy) in each cable. See Think About It section below. Sketch a force triangle and label the sides - Ftens for the hypotenuse and Fy for the vertical side. Label the angle . See Math Magic section below. Using a trigonometric function, calculate the tension force from knowledge of the angle and the Fy value. See Math Magic section below. All the individual forces acting upon the sign must balance. The cables are at an angle, so each cable has a vertical and a horizontal component of tension. Since the sign is hung symmetrically, the weight of the sign is distributed equally to each cable. Thus, the vertical component of tension is the same in each cable and equal to one-half the weight of the sign. The tension in the cable is a force vector. Vectors are represented by vector arrows. Vectors such as this one have horizontal and vertical components. The components are often represented by constructing a right triangle about the vector such that the vector is the hypotenuse of the right triangle. The components are then the legs of the right triangle. Trigonometric functions can be used to relate the values of the components to the value of the vector. The legendary SOH CAH TOA is applied to the force triangle in this question to give the following results. Fx = Ftens • cosine Fy = Ftens • sine where = angle between the cable and the horizontal.

 A light fixture which weighs 82.2 N is being hung symmetrically by three cables (or four cables) which make an angle of 67.1 degrees with the horizontal. Draw a free-body diagram and perform a trigonometric analysis to determine the tension in one of the cables. (Note: Your numbers are selected at random and likely different from the numbers listed here.) Success at a problem in physics is dependent upon a carefully plotted strategy. The strategy below will prove useful in this question: Construct a free-body diagram for the sign. Represent each of the forces by vector arrows which point in the direction of each force; label the forces according to their type. Determine the vertical component of the tension (Fy) in each cable. See Think About It section below. Sketch a force triangle and label the sides - Ftens for the hypotenuse and Fy for the vertical side. Label the angle . See Math Magic section below. Using a trigonometric function, calculate the tension force from knowledge of the angle and the Fy value. See Math Magic section below. All the individual forces acting upon the sign must balance. The cables are at an angle, so each cable has a vertical and a horizontal component of tension. Since the sign is hung symmetrically, the weight of the sign is distributed equally to each cable. Thus, the vertical component of tension is the same in each cable. If there are three cables, the vertical component in each one is equal to one-third the weight of the sign. If there are four cables, the vertical component in each one is equal to one-fourth the weight of the sign. The tension in the cable is a force vector. Vectors are represented by vector arrows. Vectors such as this one have horizontal and vertical components. The components are often represented by constructing a right triangle about the vector such that the vector is the hypotenuse of the right triangle. The components are then the legs of the right triangle. Trigonometric functions can be used to relate the values of the components to the value of the vector. The legendary SOH CAH TOA is applied to the force triangle in this question to give the following results. Fx = Ftens • cosine Fy = Ftens • sine where = angle between the cable and the horizontal.