A crate is supported by two ropes. One rope makes an angle of 46^∘ 20^' with the horizontal and

A crate is supported by two ropes. One rope makes an angle...

A crate is supported by two ropes. One rope makes an angle of $46^{\circ} 20^{\prime}$ with the horizontal and has a tension of 89.6 pounds on it. The other rope is horizontal. Find the weight of the crate and the tension in the horizontal rope.

Key Concepts

Trigonometric Functions

Trigonometric functions, particularly sine and cosine, are crucial in calculating the components of forces that act at an angle. These functions relate the angles with the force magnitudes, enabling accurate decomposition of forces in problems involving inclined angles.

Static Equilibrium

Static equilibrium refers to the condition where an object remains at rest because the net force acting on it is zero. This means the sum of all forces in both the horizontal and vertical directions must cancel out, forming the basis for setting up the equilibrium equations.

Free-Body Diagram

A free-body diagram is a simplified diagram that depicts all the external forces acting on an object. It is essential for visualizing the problem setup and forming the equations needed to analyze the forces in equilibrium.

Resolution of Forces

Resolution of forces involves breaking a force into its horizontal and vertical components. This technique uses trigonometric functions such as sine and cosine, allowing one to analyze the effects of each component separately in equilibrium problems.

Transcript

00:01 You have a question which is 93 number question in which there is a crate which is supported by the two ropes okay if this is okay this is a horizontal construction and the crate is here so it is given that one rope is making an angle 46 point sorry 46 degrees if this is theta theta is given as 46 degrees and 20 minutes and the tension is 89 point if this is t1 so tension is being given as 189 .6 pounds okay the other rope is horizontal other rope is horizontal so let us suppose this is the other rope which is horizontal okay and let us suppose here it is tension t2 so we have to find the weight of the crate and tension in the horizontal rope so if this is w we have to find the weight and t2 okay so let us just uh just break this t1 in two components that is one will be horizontal and other will be the vertical component this will be t1 or sine theta sine theta and this horizontal component will be t1 cost theta okay now since this is in equilibrium so t1 cos theta will balance t2 and t1 sine theta will balance w so first of all let us find out weight weight will be equal to t1 sine theta so t1 is being given as 189 .6 sine theta sine theta sine 46 degrees and 20 minutes okay so 189 point six into sign now 20 minutes will be equal to 20 by 60 degrees so one by three degrees so we can say that this 46 degree we should write 46 .333 degrees so this will be 89 .6 .3 degrees into sign 46 .333 degrees so this will be 89 .6 into sign 46 .333 degrees so 64 .81 okay so 64 .81 pounds so weight will be equal to 64 .81 pounds now b we have to find t2 now balancing t1 cost theta so t2 will be simply equal to t2 will be equal to t1 cost theta now t1 is being given 89 .6 pounds and cos 46 .3383 degrees okay so 89 .6 into cost 46 .333 degrees which is 61 .86.

03:27 This is 61 .86 is this correct? 89 .6 into cost 46.

03:37 336.

03:38 63...

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