A 0.16-g spider hangs from the middle of the first thread of its future web. The thread makes an angle 7.2* with the horizontal on both sides of the spider. (A) What is the tension in the thread? (B) If the angle made by the thread had been less than 7.2*, would its tension have been greater than, less than, or the same as in part (A)? Explain.
Let T be the tension in both threads. (A horizontal force balance will show that T is the same on both sides if the angles are the same).
A vertical force balance requires that:
(A)
2 T sin7.2 = M g
= 1.6*10^-4 kg * 9.8 m/s^2
= 1.57*10^-3 Newtons
T = 1.57*10^-3/(2 sin7.2) = ____ N
(B) What happens to T when the angle changes?
(The sine of an angle between 0 and 90 degrees increases with the angle.)
To find the tension in the thread, we can use the concept of equilibrium. When an object is in equilibrium, the net force acting on it is zero. In this case, the forces acting on the spider are the tension in the thread and the gravitational force.
(A) To find the tension in the thread:
1. Draw a force diagram for the spider.
- Label the downward force due to gravity as "mg" (mass * acceleration due to gravity).
- Label the tension in the thread as "T".
- The angle between the thread and the horizontal is 7.2 degrees on both sides, so we have two forces: T and T, both at an angle of 7.2 degrees with the horizontal.
2. Break down the weight force and tension forces into their horizontal and vertical components.
- The vertical component of the weight force is mg * sin(7.2°).
- The horizontal component of the weight force is mg * cos(7.2°).
- The horizontal component of each tension force is T * cos(7.2°).
- The vertical component of each tension force is T * sin(7.2°).
3. Write down the equations for equilibrium in the horizontal and vertical directions separately.
- In the horizontal direction: T * cos(7.2°) + T * cos(7.2°) = 0 (since the net horizontal force is zero).
- In the vertical direction: 2 * T * sin(7.2°) - mg * sin(7.2°) = 0 (since the net vertical force is zero).
4. Solve the equations simultaneously for T.
- From the first equation: 2 * T * cos(7.2°) = 0. ---> T * cos(7.2°) = 0
- From the second equation: 2 * T * sin(7.2°) - mg * sin(7.2°) = 0. ---> 2 * T * sin(7.2°) - mg * sin(7.2°) = 0. ---> 2 * T = mg
- T = mg / 2 * sin(7.2°)
Now we can substitute the values to calculate the tension in the thread:
- m = 0.16 g = 0.16 * 10^(-3) kg (since g = 9.8 m/s^2)
- θ = 7.2°
After substituting these values, you can calculate the tension in the thread.
(B) If the angle made by the thread had been less than 7.2°, the tension would be greater.
- This can be explained by considering the vertical component of the tension force. The vertical component of the tension force counteracts the weight of the spider, keeping it suspended.
- As the angle gets smaller, the vertical component of the tension force increases. This means that the tension in the thread needs to increase in order to balance the gravitational force and keep the spider in equilibrium.
- Therefore, if the angle had been less than 7.2°, the tension in the thread would have been greater in order to maintain the equilibrium condition.