The leg and cast in the figure below weigh 320 N (w1). 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 solve this problem, we need to analyze the forces acting on the hip joint. We have two forces: the weight of the leg and cast, denoted as w1, and the weight we need to find, denoted as w2.
Let's break down the weight of the leg and cast into its vertical and horizontal components. The vertical component acts downwards and is equal to w1. The horizontal component acts towards the hip joint. This horizontal component can be found using basic trigonometry.
To find the weight w2 and the angle 𝛼, we need to set up an equilibrium condition where no net force is exerted on the hip joint. In other words, the vertical and horizontal forces must balance each other.
Here's how we can proceed:
1. Resolve the weight of the leg and cast, w1, into its vertical and horizontal components.
- The vertical component is w1 * sin(𝛼).
- The horizontal component is w1 * cos(𝛼).
2. Write down the equilibrium condition for the vertical and horizontal forces acting on the hip joint.
- Vertical equilibrium: the sum of the upward and downward forces is equal to zero.
- Upward force: this is the weight w2.
- Downward force: this is the vertical component of w1, which is w1 * sin(𝛼).
- So, we have w2 - w1 * sin(𝛼) = 0.
- Horizontal equilibrium: the sum of the forces in the left and right directions is equal to zero.
- Right force: this is the horizontal component of w1, which is w1 * cos(𝛼).
- Left force: there is no force acting towards the left direction.
- So, we have w1 * cos(𝛼) = 0.
3. Solve the equations for w2 and 𝛼.
- From the vertical equilibrium equation, we can solve for w2 as w2 = w1 * sin(𝛼).
- From the horizontal equilibrium equation, we can see that there is no force acting towards the left, which means the equation is always satisfied.
- Therefore, we only need to solve for 𝛼 by setting up the equation w2 = w1 * sin(𝛼).
4. Substitute the value of w1 (320 N) into the equation.
- w2 = 320 * sin(𝛼).
Now we can use this equation to find the weight w2 and the angle 𝛼 needed so that no force is exerted on the hip joint by the leg plus cast.