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Question Find the tension in two ropes supporting a 400N400 \mathrm{N} object, where one rope makes a 40∘40^{\circ} angle with the ceiling and the other makes a 50∘50^{\circ} angle with the wall.

Studdy Solution

STEP 1

Assumptions1. The weight of the object is400 N. . The angle between the first rope and the ceiling is40 degrees.
3. The angle between the second rope and the wall is50 degrees.
4. The forces are in equilibrium, meaning the sum of the forces is zero.

STEP 2

a) Drawing a vector diagram is a visual task and not possible to illustrate in LaTeX. However, we can describe how to draw it. The object is at the center, with the weight vector pointing downwards. The tension in the first rope is represented by a vector pointing towards the ceiling at an angle of40 degrees. The tension in the second rope is represented by a vector pointing towards the wall at an angle of50 degrees.

STEP 3

b) To calculate the tension in the ropes, we need to set up two equations based on the equilibrium of forces. The sum of the vertical forces must be zero, and the sum of the horizontal forces must also be zero.
For the vertical forces, we have1cos⁡(40∘)−2cos⁡(50∘)=400 N1 \cos(40^{\circ}) -2 \cos(50^{\circ}) =400 \, NFor the horizontal forces, we have1sin⁡(40∘)+2sin⁡(50∘)=01 \sin(40^{\circ}) +2 \sin(50^{\circ}) =0

STEP 4

We can solve these two equations simultaneously to find the values of 11 and 22. However, it's easier to solve if we express one variable in terms of the other. From the equation for the horizontal forces, we can express 11 in terms of 221=−2sin⁡(50∘)sin⁡(40∘)1 = -2 \frac{\sin(50^{\circ})}{\sin(40^{\circ})}

STEP 5

Substitute this expression for 11 into the equation for the vertical forces−2sin⁡(50∘)sin⁡(40∘)cos⁡(40∘)−2cos⁡(50∘)=400 N-2 \frac{\sin(50^{\circ})}{\sin(40^{\circ})} \cos(40^{\circ}) -2 \cos(50^{\circ}) =400 \, N

STEP 6

olve this equation for 222=400 N−sin⁡(50∘)sin⁡(40∘)cos⁡(40∘)−cos⁡(50∘)2 = \frac{400 \, N}{-\frac{\sin(50^{\circ})}{\sin(40^{\circ})} \cos(40^{\circ}) - \cos(50^{\circ})}

STEP 7

Calculate the value of 22.

STEP 8

Substitute the value of 22 into the equation for 11 to find the value of 111=−2sin⁡(50∘)sin⁡(40∘)1 = -2 \frac{\sin(50^{\circ})}{\sin(40^{\circ})}

STEP 9

Calculate the value of $$.
The values of $$ and $2$ are the tensions in the two ropes.

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