Question

A loudspeaker of mass 23.0kg is suspended a distance of h = 1.80m below the ceiling by two cables that make equal angles with the ceiling. Each cable has a length of l = 3.90m.
What is the tension T in each of the cables?
Use 9.80 m/s^2 for the magnitude of the acceleration due to gravity.

Answer 1

Answer:

T = 243.80 N

Explanation:

In order to calculate the tension T in each of the cables, you take into account the Newton second law for the vertical component of the forces involved.

$Tsin\theta+Tsin\theta-W=0$     (1)

where you have used the fact that both cables have the same tension T, and they make the same angle with the horizontal.

W is the weigh of the loudspeaker and has the folowing values:

$W=Mg=(23.0kg)(9.8m/s^2)=225.4N$

Next, you can calculate the angle θ by using the information about the length of the cables and the distance from the ceiling to the loudspeaker:

$sin\theta=\frac{1.80m}{3.90m}=0.461\\\\\theta=sin^{-1}(0.461)=27.48\°$

Finally, you solve the equation (1) for T, and then you replace the values of the other parameters:

$2Tsin\theta-W=0\\\\T=\frac{W}{2sin\theta}=\frac{225.4N}{2sin(27.48\°)}=243.80N$

The tension in each cable is 243.80N

Answer 2

The tension (T) in each of the cables is equal to 244.07 Newton.

Given the following data:

• Mass = 23.0 kg

• Height, h = 1.80 m

• Length, l = 3.90 m.

• Acceleration due to gravity = 9.8 $m/s^2$

To find the tension (T) in each of the cables:

First of all, we would determine the weight of the loudspeaker.

$Weight = mg\\\\Weight = 23.0 \times 9.8$

Weight = 225.4 Newton

Next, we would apply Newton's Second Law of Motion to find the tension (T) in each of the cables:

$TSin\theta + TSin\theta - Weight = 0\\\\2TSin\theta-Weight = 0\\\\2TSin\theta = Weight\\\\T = \frac{Weight}{2Sin\theta}$...equation 1.

But, $Sin\theta = \frac{h}{l}$

$Sin\theta = \frac{1.8}{3.9} \\\\Sin\theta =0.4615\\\\\theta = Sin^{-1}(0.4615)\\\\\theta =27.5^\circ$

Substituting the parameters into eqn. 1, we have;

$T = \frac{225.4}{2 \times sin(27.5)} \\\\T = \frac{225.4}{0.9235}$

T = 244.07 Newton

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