The leg and cast in the figure below weigh 260 N, with the center of mass as indicated by the blue arrow in the diagram. The counterbalance w1 weighs 125 N. Determine the weight w2 AND the angle α needed so that no force is exerted on the hip joint by the leg plus cast.

To determine the weight w2 and the angle α needed so that no force is exerted on the hip joint by the leg plus cast, we can set up a force balance equation.

Let's consider the forces acting on the hip joint. We have the weight of the leg and cast, which is directed downward and has a magnitude of 260 N. We also have the counterbalance force w1, which is directed upward and has a magnitude of 125 N. Finally, we have the weight w2, which is directed downward at an angle α.

Since no force is exerted on the hip joint, the net vertical force should be zero. This means the upward force from the counterbalance w1 should balance the downward force from the leg and cast weight plus w2.

In mathematical terms, we can write:

w1 - (260 N + w2) * sinα = 0

From this equation, we can solve for w2 and α.

First, let's rearrange the equation:

w1 = (260 N + w2) * sinα

Next, we can substitute the given values:

125 N = (260 N + w2) * sinα

Now, rearranging the equation, we can solve for w2:

w2 = (125 N/sinα) - 260 N

To determine the angle α, we need to use trigonometry. Rearranging the equation, we get:

sinα = (125 N - w2) / 260 N

Now we can use inverse sine (sin^-1) to get the value of α:

α = sin^-1((125 N - w2) / 260 N)

By substituting the value of w2 into this equation, we can calculate the angle α.