Consider the 667 N weight held by two cables shown below. The left-hand cable had tension 890 N and makes an angle of θ with the wall. The right-hand cable had tension 960 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? Round up to four significant digits. Express unit as ”degrees”.
b) What is the angle θ which the left-hand cable makes with respect to the wall? Round up to four significant digits. Express unit as ”degrees”.
The process has stumbled me for a while, but am certain that I need to use sin^2 + cos^2=1 somewhere along the way. I appreciate your help.
To find the angle θ1 that the right-hand cable makes with the ceiling, we can use trigonometry. Let's call the vertical component of the tension in the right-hand cable T1y.
Using the given values, we can use sin θ1 = T1y / T1 to find the sine of the angle θ1, where T1 is the tension in the right-hand cable.
sin θ1 = T1y / T1
To find T1y, we can use the tension T1 and the angle θ1 with the ceiling. The vertical component of T1, T1y, can be found using the formula T1y = T1 * sin θ1.
Now, let's substitute T1y into the equation:
sin θ1 = (T1 * sin θ1) / T1
Canceling out T1:
sin θ1 = sin θ1
This equation is true for all values of θ1. Therefore, we cannot determine the value of θ1 accurately using this information alone.
However, to find the angle θ that the left-hand cable makes with the wall, we can use the same approach.
Let's call the horizontal component of the tension in the left-hand cable T2x.
sin θ = T2x / T2
To find T2x, we can use the tension T2 and the angle θ with the wall. The horizontal component of T2, T2x, can be found using the formula T2x = T2 * cos θ.
Now, let's substitute T2x into the equation:
sin θ = (T2 * cos θ) / T2
Canceling out T2:
sin θ = cos θ
Since sin θ = cos (90° - θ), we can simplify the equation:
sin θ = sin (90° - θ)
This equation is true when θ = 45° or 135°, but since the tension T2 is greater than the weight of the object, the angle θ must be less than 90°. Therefore, the angle θ that the left-hand cable makes with the wall is approximately 45°.
In summary:
a) We cannot determine the exact value of θ1 with the given information.
b) The angle θ that the left-hand cable makes with the wall is approximately 45°.