A loudspeaker of mass 20.0kg is suspended a distance of h = 1.60m below the ceiling by two cables that make equal angles with the ceiling. Each cable has a length of l = 4.00m .

To solve this problem, we need to calculate the tension in each cable.

First, let's draw a diagram of the situation:

/|
/ |
/ |
/ |
/ θ |
Tension > /____|
Ceiling

h

We know that the loudspeaker has a mass of 20.0 kg and is suspended a distance h = 1.60 m below the ceiling. The two cables have equal lengths l = 4.00 m.

Now, let's analyze the forces acting on the loudspeaker.

1. The weight of the loudspeaker: The weight of an object is given by the formula w = mg, where m is the mass and g is the acceleration due to gravity. In this case, the weight of the loudspeaker is w = (20.0 kg)(9.8 m/s^2) = 196 N. This force acts downwards.

2. Tension in the cables: Since the loudspeaker is in equilibrium (not moving), the sum of the forces in the y-direction (vertical direction) must equal zero. The tension in each cable contributes to the upward force that balances the weight of the loudspeaker.

Let's solve for the tension in each cable:

Since the two cables make equal angles with the ceiling, the vertical component of tension in each cable cancels each other out, leaving only the horizontal component that opposes the weight of the loudspeaker.

The horizontal component of each tension force can be calculated using the formula T*cos(θ), where T is the tension force and θ is the angle between the cable and the ceiling.

In this case, we have two equal angles, so we can represent the horizontal component of each tension force as T*cos(θ).

Next, we can equate the sum of the horizontal components of the tension forces to the weight of the loudspeaker:

2*(T*cos(θ)) = 196 N

Now, let's calculate the angle (θ):

Using the given information, we know that the height (h) and the distance (l) form a right-angled triangle.

Using trigonometry, we can find the angle θ:

sin(θ) = h/l
sin(θ) = 1.60 m / 4.00 m
θ = arcsin(0.40) ≈ 23.6 degrees

Now we can substitute the value of θ into our equation:

2*(T*cos(23.6 degrees)) = 196 N

Finally, we can solve for the tension force (T):

T = 196 N / (2*cos(23.6 degrees))

Using a calculator, we find:

T ≈ 109.5 N

Therefore, the tension in each cable is approximately 109.5 N.