1.81 Muscles and Bones: Vector Magnitude Problem

[cdlegendary]

Homework Statement

A patient in therapy has a forearm that weighs 23.0 N and that lifts a 114.0 N weight. These two forces have direction vertically downward. The only other significant forces on his forearm come from the biceps muscle (which acts perpendicularly to the forearm) and the force at the elbow. If the biceps produces a pull of 232 N when the forearm is raised 44 degrees above the horizontal, find the magnitude and direction of the force that the elbow exerts on the forearm. (The sum of the elbow force and the biceps force must balance the weight of the arm and the weight it is carrying, so their vector sum must be 137.0 N, upward.)

Homework Equations

So there's 3 vectors I think:
A: angle ?, magnitude -137N
B: angle 44 degrees, magnitude ?
C: angle 134 degrees, magnitude 232 N

The Attempt at a Solution

I don't think I have my question setup correctly, but I tried:

232sin(44) = 161.2N = x
232cos(44) = 166.9N = y

Then I use the law of cosines, but doesn't really work..if someone could walk me through the setup, I would greatly appreciate it.

 

[ideasrule]

First, draw a free-body diagram of the forearm. Label all forces, then write out Newton's second law for both the x and y directions.

 

[kerimek]

As a scientist, it is important to approach problems like this with a clear understanding of the concepts and principles involved. Let's break down the problem and apply the relevant equations to find a solution.

First, we need to identify the forces acting on the forearm. These include the weight of the forearm (23.0 N), the weight it is lifting (114.0 N), the force from the biceps muscle (232 N), and the force from the elbow (unknown).

Next, we need to consider the direction of these forces. The weight of the forearm and the weight it is lifting are both acting vertically downward. The force from the biceps muscle is acting perpendicularly to the forearm, and the force from the elbow is acting at an angle of 134 degrees (180-44) from the horizontal.

To find the magnitude and direction of the force from the elbow, we can use the principle of vector addition. Since the vector sum of all the forces must be equal to 137.0 N, we can set up an equation:

23.0 N + 114.0 N + 232 N + F = 137.0 N

Solving for F, we get:

F = -232 N (downward)

This means that the force from the elbow must be 232 N in magnitude and act downward to balance out the other forces and keep the forearm in equilibrium.

To find the angle of this force, we can use the law of cosines:

F^2 = 232^2 + 137^2 - 2(232)(137)cosθ

Solving for θ, we get:

θ = 144.5 degrees

Therefore, the force from the elbow is 232 N acting downward at an angle of 144.5 degrees from the horizontal.

In summary, by understanding the forces involved and applying the principles of vector addition and the law of cosines, we can solve this problem and determine the magnitude and direction of the force from the elbow on the forearm. This approach is crucial for any scientist, as it allows for a logical and systematic way of solving complex problems.