A weight of 690 N is held by two cables. The left-hand cable had tension T2 and makes an angle of 41◦ with the ceiling. The right-hand cable had tension T1 and makes an angle of 42◦ with the ceiling.
What is the tension in the cable labeled T1 slanted at an angle of 42◦?
To find the tension in the cable labeled T1 slanted at an angle of 42°, we'll use the vertical component of the force equation.
First, let's find the vertical component of the weight of 690 N that is acting downward. The weight can be expressed as:
Weight = mass x gravitational acceleration
Since weight is equal to 690 N, we can write:
690 = mass x 9.8
To solve for the mass, we divide both sides by 9.8:
mass = 690 / 9.8 ≈ 70.408 kg
Now, let's find the vertical component due to the tension in the left-hand cable (T2). We'll use the angle of 41° to calculate this:
Vertical component of T2 = T2 x sin(41°)
Next, let's find the vertical component due to the tension in the right-hand cable (T1), which is the tension we're looking for:
Vertical component of T1 = T1 x sin(42°)
The total vertical force acting upward can be found by adding the vertical components together:
Total vertical force = Weight + Vertical component of T2 + Vertical component of T1
Since the total vertical force is balanced (net force is zero as the system is in equilibrium), it must equal zero. Therefore, we can write:
0 = Weight + Vertical component of T2 + Vertical component of T1
Plugging in the values we calculated:
0 = 690 + T2 x sin(41°) + T1 x sin(42°)
Now we can solve for T1:
T1 x sin(42°) = - (Weight + T2 x sin(41°))
Dividing both sides by sin(42°), we have:
T1 = - (Weight + T2 x sin(41°)) / sin(42°)
Substituting the known values:
T1 = - (690 + T2 x sin(41°)) / sin(42°)
Please note that because the weight is acting downward and the tensions in the cables are acting upward, the tension calculated may have a negative sign indicating opposite direction to weight.
By substituting the value of T2 and solving the equation, we can find the tension in the cable labeled T1 slanted at an angle of 42°.