You are holding a ball of mass m2 in your hand. In this problem you will solve for the upward force T⃗ that the tendon of your biceps muscle exerts to keep the forearm horizontal and the downward force F⃗ that the upper arm exerts on the forearm at the elbow joint. Assume the outstretched arm has a mass of m1 , the center of mass of the outstretched arm is a distance s from the elbow, the tendon attaches to the bone a distance d from the elbow, and the ball is a distance 2s from the elbow. (Taking T⃗ to be upward and F⃗ to be downward, with no horizontal components, indicates that this is a simplified model. ) A schematic representation of this situation is shown below:

Hint: The forces can be modeled as shown in the following Free Body Diagram:

(a) What is the magnitude of the tension T≡∣∣T⃗ ∣∣ in the tendon? Express your answer in terms of s, m1, m2, d and g as needed.

T=

(b) What is the magnitude of the force that the upper arm exerts on the forearm at the elbow joint? Express your answer in terms of s, m1, m2, d and g as needed.

F=

T=s*g*(m_1+2*m_2)/d

F=-(m_1+m_2)*g+s*g*(m_1+2*m_2)/d

To solve this problem, we can apply Newton's second law of motion, which states that the sum of the forces acting on an object is equal to the product of its mass and acceleration.

In this case, the forearm is being held horizontally, so the net force acting on it in the vertical direction must be zero.

Let's analyze the forces acting on the forearm:

1. Force of gravity on the forearm: The downward force due to gravity can be calculated as m1 * g, where m1 is the mass of the forearm and g is the acceleration due to gravity.

2. Force of gravity on the ball: The downward force due to gravity acting on the ball can be calculated as m2 * g, where m2 is the mass of the ball.

3. Tension in the tendon: The tendon exerts an upward force (T) to counterbalance the downward forces on the forearm and the ball.

4. Force exerted by the upper arm: The upper arm exerts a downward force (F) on the forearm at the elbow joint, as indicated in the problem.

Since the forearm is held horizontally, we can set up the following equation:

T + F - m1 * g - m2 * g = 0

From the given information, we can express the distances as follows:
- The tendon attaches to the bone a distance d from the elbow.
- The center of mass of the outstretched arm is a distance s from the elbow.
- The ball is a distance 2s from the elbow.

Now, let's solve for T and F:

(a) To find the tension T in the tendon, we need to isolate T in the equation:

T = m1 * g + m2 * g - F

(b) To find the force F exerted by the upper arm, we need to isolate F in the equation:

F = m1 * g + m2 * g - T

So, the answers for (a) and (b) are:

(a) T = m1 * g + m2 * g - F
(b) F = m1 * g + m2 * g - T

Please note that these answers are expressed in terms of s, m1, m2, d, and g as mentioned in the problem.