The 636 N weight held by two cables. The left-hand cable had tension 840 N and makes an angle of θ with the wall. The right-hand cable had tension 890 N and makes an angle of θ1 with the ceiling.
a) What is the angle θ1 which the right-hand cable makes with respect to the ceiling?
b) What is the angle θ which the left-hand cable makes with respect to the wall?
74 degrees
To find the angles, θ1 and θ, in this scenario, we can use trigonometric ratios and the given information.
a) To find the angle θ1 which the right-hand cable makes with respect to the ceiling, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side in a right-angled triangle.
In this case, the opposite side is the vertical component of the tension in the right-hand cable, which is 890 N, and the adjacent side is the horizontal component of the tension in the right-hand cable. We can find the horizontal component using the cosine function.
The horizontal component of the tension in the right-hand cable is found by multiplying the tension (890 N) by the cosine of the angle θ1. So, the equation becomes:
Horizontal component = Tension * cos(θ1)
Now, we can use the tangent function to find θ1:
tan(θ1) = opposite side / adjacent side = Tension * sin(θ1) / Tension * cos(θ1) = sin(θ1) / cos(θ1)
Simplifying the equation gives us:
tan(θ1) = sin(θ1) / cos(θ1)
To solve for θ1, we can take the inverse tangent (arctan) of both sides:
θ1 = arctan(sin(θ1) / cos(θ1))
Using this equation, you can find the value of θ1.
b) Similarly, to find the angle θ which the left-hand cable makes with respect to the wall, we can use the same approach.
The horizontal component of the tension in the left-hand cable can be found using the cosine function, just like before. So, the equation becomes:
Horizontal component = Tension * cos(θ)
Now, we can use the tangent function to find θ:
tan(θ) = opposite side / adjacent side = Tension * sin(θ) / Tension * cos(θ) = sin(θ) / cos(θ)
Simplifying the equation gives us:
tan(θ) = sin(θ) / cos(θ)
To solve for θ, we can take the inverse tangent (arctan) of both sides:
θ = arctan(sin(θ) / cos(θ))
Using this equation, you can find the value of θ.