Robin is making a mobile to hang over her baby sister's crib. She purchased four stuffed animals: a teddy bear (16 g), a lamb (18 g), a little pony (22 g) and a bird (15 g). She also purchased three small wooden dowels, each 14 cm long and of mass 7 g, and thread of negligible mass. She wants to hang the bear and the pony from the ends of the first dowel and the lamb and the bird from the ends of the second dowel. Then, she wants to suspend the two dowels from the ends of the middle dowel and hang the whole assembly from the ceiling. Find where the middle thread should be attached to the middle dowel and where the ends of the middle dowel should be attached to the first and second dowel so that the entire assembly will hang level.

Question

Robin is making a mobile to hang over her baby sister's crib. She purchased four stuffed animals: a teddy bear (16 g), a lamb (18 g), a little pony (22 g) and a bird (15 g). She also purchased three small wooden dowels, each 14 cm long and of mass 7 g, and thread of negligible mass. She wants to hang the bear and the pony from the ends of the first dowel and the lamb and the bird from the ends of the second dowel. Then, she wants to suspend the two dowels from the ends of the middle dowel and hang the whole assembly from the ceiling. Find where the middle thread should be attached to the middle dowel and where the ends of the middle dowel should be attached to the first and second dowel so that the entire assembly will hang level.

Answer 1

To solve this problem, we need to consider the moments of the forces acting on each dowel. The assembly will hang level when the total torque about any axis is zero.

Let's denote the mass of the teddy bear as m1 (16 g), the mass of the lamb as m2 (18 g), the mass of the little pony as m3 (22 g), and the mass of the bird as m4 (15 g).

First, let's find the center of mass for each set of animals hanging from the first and second dowels:

For the first dowel:
The center of mass for the bear and pony set will be at a distance of x1 from the left end of the dowel, where x1 denotes the position of the middle thread on the first dowel. The mass of the bear and pony combined is (m1 + m3). Since the dowel is of length 14 cm, the distance from the left end to the center of mass is 7 cm. Therefore, we can use the formula for the center of mass:

x1 * (m1 + m3) = 7 * (m1 + m3)

Similarly, for the second dowel:
The center of mass for the lamb and bird set will be at a distance of x2 from the left end of the dowel, where x2 denotes the position of the middle thread on the second dowel. The mass of the lamb and bird combined is (m2 + m4). Since the dowel is of length 14 cm, the distance from the left end to the center of mass is 7 cm. Therefore, we get:

x2 * (m2 + m4) = 7 * (m2 + m4)

Finally, let's position the middle dowel. The total torque acting on the middle dowel will be zero when the sum of the torques from the first dowel and second dowel is zero.

The torque from the first dowel depends on the mass on that side and the distance from the middle dowel to the first dowel. Denoting this distance as d1, the torque from the first dowel is:

d1 * (m1 + m3)

Similarly, the torque from the second dowel depends on the mass on that side and the distance from the middle dowel to the second dowel. Denoting this distance as d2, the torque from the second dowel is:

d2 * (m2 + m4)

Since the middle dowel is of length 14 cm, the distances from the middle dowel to the first and second dowels are (14 - d1) on the left and d2 on the right.

Setting up the equation for the sum of torques:

d1 * (m1 + m3) = (14 - d1) * (m2 + m4) + d2 * (m2 + m4)

Now we have three equations: one for x1, one for x2, and one for the sum of torques.

Solving these equations simultaneously will give us the values of x1, x2, d1, and d2, which will help us determine where the middle thread should be attached to the middle dowel and where the ends of the middle dowel should be attached to the first and second dowel so that the entire assembly will hang level.