An artist designs a mobile of light horizontal rods connected by vertical strings and supporting various shaped weights (see figure below). Find the magnitudes w2, w3, and w4 if w1 = 5.91 units of weight. The numerical values given in the figure all have units of length.
To find the magnitudes w2, w3, and w4 in the mobile, we can analyze the forces acting on each rod.
Let's consider each horizontal rod individually:
1. Start with the topmost rod: The only force acting on this rod is the weight w1, which is directed downwards.
2. Moving down to the second rod: This rod supports the weight w1 from the top and the weight w2 from the bottom. Therefore, the net force acting on this rod must be zero (since the rod is in equilibrium). Thus, we can write the equation:
w1 = w2
3. Proceeding to the third rod: This rod supports the weight w2 from above and the weight w3 from below. Again, the net force acting on this rod must be zero. Hence, we have:
w2 = w3
4. Finally, the fourth rod: This rod supports the weight w3 from above and the weight w4 from below. Applying the equilibrium condition, we get:
w3 = w4
Now, since we know that w1 = 5.91 (given in units of weight), we can substitute this value into the equations we derived.
From equation 2 (w1 = w2):
w2 = 5.91 units of weight
From equation 3 (w2 = w3):
w3 = 5.91 units of weight
From equation 4 (w3 = w4):
w4 = 5.91 units of weight
Hence, the magnitudes of w2, w3, and w4 are all 5.91 units of weight.