an artist designs a mobile of light horizontal rods connected by vertical strings and supporting various shaped weights. Find the magnitudes w2, w3, and w4 if w1= 1.00 unit of weight. The numerical values given in the figure all have units of length.
To find the magnitudes w2, w3, and w4, we need to analyze the equilibrium of forces acting on the mobile. The key idea is that the mobile is in equilibrium, meaning that the net force acting on each rod is zero.
Let's break down the problem step by step:
1. Draw a diagram of the mobile, labeling each rod (w1, w2, w3, w4) and the given lengths.
2. Start by analyzing the top horizontal rod. Since it is the only rod connected to a vertical string, the weight w1 will cause the string to exert a tension force T1 on the rod.
3. The tension force T1 can be resolved into horizontal and vertical components. The horizontal component of T1 will balance the force exerted by the other horizontal rods, while the vertical component will balance the weight w1.
4. Analyze the second horizontal rod (w2). Since it is parallel to the top rod, its tension force T2 will have the same magnitude but an opposite direction. The horizontal component of T2 will balance the horizontal component of T1, while the vertical component will balance the weight w2.
5. Repeat the same analysis for the third and fourth horizontal rods (w3 and w4). Identify their corresponding tension forces T3 and T4, respectively.
6. Now, set up the equilibrium equations for each rod:
- For the top rod (w1): T1 - w1 = 0 (vertical equilibrium).
- For the second rod (w2): -T2 + T1 = 0 (horizontal equilibrium) and T2 - w2 = 0 (vertical equilibrium).
- For the third rod (w3): -T3 + T2 = 0 (horizontal equilibrium) and T3 - w3 = 0 (vertical equilibrium).
- For the fourth rod (w4): -T4 + T3 = 0 (horizontal equilibrium) and T4 - w4 = 0 (vertical equilibrium).
7. Substitute T1 = w1, T2 = w2, T3 = w3, and T4 = w4 in the respective equilibrium equations.
8. Solve the system of equations to find the values of w2, w3, and w4.
By following these steps, you should be able to determine the magnitudes w2, w3, and w4, based on the given information.