Pulling the string on a bow back with a force of 28.7lbs, an archer prepares to shoot an arrow. If the archer pulls in the center of the string, and the angle between the two halves is 138 degrees, what is the tension in the string?
40.04
Consider the forces acting where the arrow is held. Perform a force balance along the direction of the arrow. F = 28.7 pulls the string and the string has tension T on both sides of the arrow, inclined 69 degrees to the horizontal.
2 T cos 69 = 28.7 lb
Solve for T
40.04 lb
Ah, the archer and the tension in the string, quite the tale! Let's get calculating, shall we?
Now, you say the archer pulls with a force of 28.7lbs in the center of the string, causing an angle of 138 degrees between the two halves. Fascinating!
To determine the tension, we need to divide the force applied by the sine of half the angle. So, let's crunch the numbers:
First, we'll find half the angle: 138 degrees / 2 = 69 degrees.
Next, we'll apply some math magic with sine: sin(69 degrees) = 0.9397.
Finally, we'll divide the applied force by the sine: 28.7lbs / 0.9397 ≈ 30.51 lbs.
Boom! The tension in the string is approximately 30.51 lbs. But hey, don't let the numbers string you along too much, stay on target and keep having fun!
To find the tension in the string, we can use the concept of equilibrium for forces. In this case, we'll consider the forces acting on each half of the string.
First, we need to convert the force from pounds (lbs) to the SI unit of force, which is Newtons (N). We can use the conversion factor: 1 lb = 4.44822 N.
So, the force applied by the archer is:
Force (in Newton) = 28.7 lbs * 4.44822 N/lb = 127.17 N
Since the archer is pulling at the center of the string, the force is divided equally between the two halves. Thus, each half of the string experiences a force of 127.17 N / 2 = 63.585 N.
The angle between the two halves of the string is given as 138 degrees. Since we have a triangle, we can use trigonometry to calculate the tension in the string.
Using the law of sines, we have:
Tension / sin(angle) = Force / sin(180 - angle)
Substituting the given values, we get:
Tension / sin(138 degrees) = 63.585 N / sin(180 - 138 degrees)
Now we can solve for the tension in the string:
Tension = sin(138 degrees) * (63.585 N / sin(180 - 138 degrees))
Calculating this expression, we find the tension in the string:
Tension ≈ 59.16 N (rounded to two decimal places)
Therefore, the tension in the string is approximately 59.16 Newtons.