A 50- lb weight is hung by a cable so that the two portions of the cable make angles of 40^∘ and

A 50- lb weight is hung by a cable so that the two portions...

Key Concepts

Systems of Equations in Statics

In statics problems, the equilibrium conditions lead to a set of simultaneous equations, one for each direction (usually horizontal and vertical). Solving these equations together is a systematic approach to finding unknown forces, such as tensions or components thereof.

Free-Body Diagrams

A free-body diagram is a visual representation that isolates an object and illustrates all the applied forces. Drawing such a diagram helps in identifying the different forces and their directions, which is critical for setting up the necessary equilibrium equations.

Static Equilibrium

Static equilibrium refers to the condition where an object remains at rest because the sum of all forces acting on it is zero. This principle is used to set up equations for both the horizontal and vertical components of force, ensuring that everything balances out and no net force causes acceleration.

Force Resolution into Components

Resolving forces into components involves breaking down a force vector into its horizontal and vertical parts using trigonometric functions. This technique is crucial for analyzing problems where forces act at angles, as it allows for the independent application of the equilibrium conditions in each direction.

Trigonometric Relationships in Mechanics

Trigonometric functions such as sine and cosine are used to connect the angles at which forces act with the magnitudes of their horizontal and vertical components. These relationships are essential in determining how much of a force contributes to equilibrium along a particular axis.

Transcript

00:01 Hi, there is a question you said that if 50 pound weight is hung by cable so that the two portions of the cable make an angle 40 degrees and 53 degrees respectively with the horizontal find the magnitudes of the forces of tensions tensions t 1 and t 2 in the cables if the resultant force acting on the object is 0 down to two decimal places okay, so this is like this yeah 50 pound weight this is 40 degree this is 53 degrees and tensions with tensions give a look at a distant forces okay let us say tension is t1 tension is t2 so if i take this x direction this is y axis and x axis so i'll be first resolving this tensions t1 and t2 in the respective rectangular components let us say this line is parallel these two are parallel so this angle is gonna be 40 40 degree that is the alternate angle that means the horizontal component of t1 will be t1 cos 40 degrees and the vertical component will be t1 sine 40 degrees similarly this is going to be equal to 53 degree so horizontal component is t2 cos 53 degrees and vertical component of t2 is t2 sine 53 degrees since it is in the resultant force acting on object is zero so forces in x direction must be equal to zero so these two forces t1 cos 40 degrees and t2 sine cos 53 degrees will just balance each other 53 degree equal to t1 cos 40 degree so t2 will be equal to t1 cos 40 degree divided by cos 53 degrees i'll be writing it as cos 40 degree divided by cos 53 degree so this is 1 .272891 equation number one the vertical component only weight and sum of these two will become zero for the equilibrium and the vertical component or in the vertical direction so t1 sine 40 degree plus t2 sine 53 degree equal to 50 so value this is t1 sine 40 degree and value of t2 i'll be plugging in 1 .27289 multiplied by t2 sine 53 degree equal to 50 okay t2 we have already put it here value of t2a like t1 yes now t1 will be taken as common sine 40 degree plus 1 .2789 nine and two seven two eight nine sine 53 degree equal to 50 therefore t1 let us solve it up sine 53 degree plus sine 40 degree this is one point six five nine three six two nine equal to 50 so t1 equal to 50 divided by one point six 6593 629 50 divided by 1 .6593 6593 629 30 .13204646 now from equation number 1 t2 is 1 .27289 1 .27289 into t1 30 into 13204646 multiplied by 1 .27289 38 .3547 and the forces are or tensions are t1 approximately 30 .13 pounds and t2 approximately 38 .35 pounds...

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