A man holds a 185-N ball in his hand, with the forearm horizontal (see the figure). He can support the ball in this position because of the flexor muscle force , which is applied perpendicular to the forearm. The forearm weighs 24.3 N and has a center of gravity as indicated. Find (a) the magnitude of and the (b) magnitude and (c) direction (as a positive angle counterclockwise from horizontal) of the force applied by the upper arm bone to the forearm at the elbow joint.
May I please get help with this question. I'm very confused. I appreciate all the assistance.
To solve this problem, we can apply the principle of equilibrium for the forearm to find the force exerted by the upper arm bone at the elbow joint.
First, let's set up the problem:
- Let Fm be the flexor muscle force
- Let Fe be the force exerted by the upper arm bone at the elbow joint
- Let Fa be the force applied by the upper arm bone to the forearm at the elbow joint
- Let d1 be the distance between the elbow joint and the center of gravity of the forearm (assume d1 = 0.17 m, as this is roughly the midpoint of an average human forearm)
- Let d2 be the distance between the elbow joint and the center of gravity of the ball (assume d2 = 0.4 m for this problem)
- Let d3 be the distance between the elbow joint and the point where the flexor muscle force is applied (assume d3 = 0.05 m)
Now, we have the following equations from equilibrium:
ΣFy = 0 => Fa - Fm - 185 = 0 (1)
ΣFx = 0 => Fe = 0 (2)
ΣM (elbow) = 0 => -Fm*d3 + 185*d2 - 24.3*d1 = 0 (3)
From equation (2) we can say that there is no horizontal force at the elbow joint exerted by the upper arm bone, therefore the force is purely vertical.
Now substituting the assumed values of d1, d2, and d3 in equation (3), we can find the value of Fm:
- Fm * 0.05 + 185 * 0.4 - 24.3 * 0.17 = 0 => Fm = (185 * 0.4 - 24.3 * 0.17) / 0.05 = 1348 N
Now, using equation (1) we can find the value of Fa:
Fa - 1348 - 185 = 0 => Fa = 1533 N
So the magnitude of the force applied by the upper arm bone to the forearm at the elbow joint is 1533 N. Since there is no horizontal component to this force, the angle is simply 0° counterclockwise from the horizontal.
To summarize:
a) Magnitude of force exerted by the upper arm bone at the elbow joint: Fe = 0 N
b) Magnitude of force applied by the upper arm bone to the forearm at the elbow joint: Fa = 1533 N
c) Direction of force exerted by the upper arm bone at the elbow joint: 0° counterclockwise from the horizontal
To solve this problem, we need to consider the forces acting on the forearm. We have the weight of the ball (185 N) pulling the forearm downwards and the weight of the forearm itself (24.3 N) acting downwards at its center of gravity.
(a) Let's first find the magnitude of the flexor muscle force (F). Since the forearm is in equilibrium (not moving), the sum of the forces acting on it must be zero. Therefore, the magnitude of the flexor muscle force must be equal to the sum of the magnitudes of the weight of the ball and the weight of the forearm:
F = weight of the ball + weight of the forearm
F = 185 N + 24.3 N
F = 209.3 N
So, the magnitude of the flexor muscle force is 209.3 N.
(b) Now, let's find the magnitude of the force applied by the upper arm bone to the forearm at the elbow joint. This force will be equal and opposite to the flexor muscle force. Therefore, the magnitude of the force applied by the upper arm bone (F') is also 209.3 N.
(c) To find the direction of the force applied by the upper arm bone, we need to determine the angle counterclockwise from the horizontal. In the figure, it is indicated that the forearm is horizontal. This means that the force applied by the upper arm bone is acting vertically upward, opposite to the weight of the ball and the forearm. So, the direction of the force applied by the upper arm bone is straight up, or 90 degrees counterclockwise from the horizontal.
In summary:
(a) The magnitude of the flexor muscle force is 209.3 N.
(b) The magnitude of the force applied by the upper arm bone is 209.3 N.
(c) The direction of the force applied by the upper arm bone is 90 degrees counterclockwise from the horizontal.
To solve this problem, we need to apply Newton's second law of motion and consider the forces acting on the forearm and ball system.
Let's break down the problem step by step:
Step 1: Calculate the weight of the ball.
Given that the weight of the ball is 185 N, this force is acting downward from the center of gravity of the ball.
Step 2: Calculate the weight of the forearm.
Given that the weight of the forearm is 24.3 N, this force is acting downward at the center of gravity of the forearm.
Step 3: Calculate the vertical force acting on the forearm.
The man can support the ball because of the flexor muscle force, which is perpendicular to the forearm. This vertical force cancels out the vertical component of the weight of the ball and the forearm together.
Step 4: Calculate the horizontal force acting on the forearm.
The horizontal force acting on the forearm is the force applied by the upper arm bone at the elbow joint.
Now, let's calculate the answers to the questions:
(a) Magnitude of the force applied by the upper arm bone to the forearm at the elbow joint:
Since the forearm is in a horizontal position, the force applied by the upper arm bone must balance out the horizontal component of the weight of the ball and the forearm.
(b) Magnitude of the force applied by the upper arm bone to the forearm at the elbow joint:
Since the force applied by the upper arm bone balances out the horizontal component of the weight of the ball and the forearm, it can be calculated using trigonometry.
We can use the equation: force = weight × cosine(angle), where the angle is the angle between the forearm and the horizontal direction.
(c) Direction of the force applied by the upper arm bone to the forearm at the elbow joint:
The direction can be determined by finding the angle between the forearm and the horizontal direction. This can be calculated using trigonometry.
To solve for the magnitude and direction of the force, you need to know the angle between the forearm and the horizontal direction. Unfortunately, the figure you mentioned is not provided, so I cannot determine the exact angle value.