Final answer:
These problems involve applying Newton's Laws of motion and trigonometric identities to calculate the tension in ropes or cables supporting a weight, which in these cases are a girl on a swing and a traffic light.
Step-by-step explanation:
To solve these problems involving Newton's Laws of motion and tension in ropes or cables, we need to employ both the laws of motion and trigonometric identities. Let's break down the first problem:
Problem 25:
- Draw a free-body diagram with the forces acting on the girl and the swing.
- Resolve the forces into components: the tension in the ropes (T) will have both vertical (Ty) and horizontal (Tx) components. Here, Ty has to balance the girl's weight (W = mg, where m is her mass and g is the acceleration due to gravity), and Tx is equal to the horizontal force (F) holding the swing at 30.0°.
- Use trigonometric identities to express Ty and Tx in terms of T and the angle given.
- Since the swing is at rest and the vertical forces must balance, 2Ty = W, and since the swing is being held at rest horizontally, Tx = F. From these equations, you can solve for T (the tension in each rope) and the magnitude of F using the mass of the girl and the angle of the ropes with vertical.
Problem 26:
In a similar manner, we would analyze the traffic light being supported by three cables. Create a free-body diagram, resolve weight into components along the cables' directions and solve for the tensions such that the vector sum of the tensions balances the weight of the traffic light.