A man holds a 178N ball in his hand, with the forearm horizontal. He can support the ball in this position because of the flexor muscle force M, which is applied perpendicular to the forearm. The forearm weighs 22 N and has a center of gravity as indicated. Find (A) the magnitude of M and (B) the magnitude and direction of the forceapplied by the upper arm bone to the forearm at the elbow jount
---Elbow jont
-------------|----|-----|------------|
-------------|----|-----|------------|
----0.0510m==|____|_____|=0.0890m----|
-------------|---------0.330m--------|
To solve this problem, we need to analyze the forces acting on the forearm. Let's start by drawing a free body diagram:
1. Draw the forearm as a straight line with a length of 0.33m.
2. Place the weight of the forearm (22N) at the center of gravity of the forearm, which is 0.0510m away from the elbow joint.
3. Place the weight of the ball (178N) at the same location as the center of gravity of the forearm.
Now let's solve for the unknowns:
(A) The magnitude of the flexor muscle force M:
To find the magnitude of M, we need to balance the moments (torques) about the elbow joint:
Sum of anticlockwise torques = Sum of clockwise torques
Clockwise torque:
22N * 0.330m
Anticlockwise torque:
178N * 0.0890m
M * 0.0510m
Setting up the equation:
22 * 0.330 = 178 * 0.0890 + M * 0.0510
Now solve for M:
M = (22 * 0.330 - 178 * 0.0890) / 0.0510
(B) The magnitude and direction of the force applied by the upper arm bone to the forearm at the elbow joint:
To find the magnitude and direction of this force, we need to consider the vertical forces acting on the forearm.
Vertical forces:
1. The weight of the forearm (22N) acting downward.
2. The weight of the ball (178N) acting downward.
3. The force applied by the upper arm bone at the elbow joint, acting upward.
Since the forearm is in equilibrium (not accelerating vertically), the sum of these vertical forces must be zero:
Sum of upward forces = Sum of downward forces
Upward force:
Force applied by the upper arm bone at the elbow joint.
Downward forces:
22N (weight of the forearm)
178N (weight of the ball)
Setting up the equation:
Force applied by the upper arm bone at the elbow joint = 22N + 178N
Now calculate the result:
Force applied by the upper arm bone at the elbow joint = 200N (upward)
So, the answers are:
(A) Magnitude of M = (22 * 0.330 - 178 * 0.0890) / 0.0510
(B) Magnitude and direction of the force applied by the upper arm bone at the elbow joint = 200N (upward)
To find the magnitude of the flexor muscle force (M), we can start by analyzing the torque (moment) about the elbow joint caused by the weight of the ball (178 N) and the weight of the forearm (22 N).
First, we need to calculate the torque caused by the weight of the ball about the elbow joint. Torque is calculated as the product of the force and the distance from the force to the point of rotation.
Torque due to the weight of the ball = Force x Distance
Torque_ball = 178 N x 0.0890 m
Next, we need to calculate the torque caused by the weight of the forearm about the elbow joint. Again, torque is calculated as the product of the force and the distance from the force to the point of rotation.
Torque due to the weight of the forearm = Force x Distance
Torque_forearm = 22 N x 0.330 m
The elbow joint is in equilibrium, so the sum of the torques acting on it must be zero. Therefore, the torque caused by the flexor muscle force (M) must be equal in magnitude and opposite in direction to the sum of the torques caused by the weight of the ball and forearm.
Torque_M - Torque_ball - Torque_forearm = 0
Now we can substitute the values and solve for the magnitude of M.
M x Distance_M - Torque_ball - Torque_forearm = 0
M x Distance_M = Torque_ball + Torque_forearm
M = (Torque_ball + Torque_forearm) / Distance_M
Substituting the given values:
M = (178 N x 0.0890 m + 22 N x 0.330 m) / Distance_M
Now, you can calculate the magnitude of M by substituting the given values for Distance_M and solving the equation.
To find the magnitude and direction of the force applied by the upper arm bone to the forearm at the elbow joint, we can use a similar approach.
Since the elbow joint is in equilibrium, the sum of the vertical forces acting on the joint must be zero. This means that the force applied by the upper arm bone must be equal in magnitude and opposite in direction to the sum of the weights of the ball and forearm.
Force_applied - Weight_ball - Weight_forearm = 0
Force_applied = Weight_ball + Weight_forearm
Force_applied = 178 N + 22 N
The magnitude of the force applied by the upper arm bone to the forearm at the elbow joint is the sum of the weights of the ball and forearm.
To find the direction of the force, we can determine whether it is upward or downward based on the equilibrium condition. If the sum of the vertical forces is positive, the force is upward. If the sum of the vertical forces is negative, the force is downward. In this case, since the weight of the ball and forearm are positive (downward), the force applied by the upper arm bone must be upward to balance the forces. Hence, the direction of the force is upward.
The problem is worked out on page 10 of
http://www.phys.lsu.edu/classes/fall2007/phys2001-24/lecture19.pdf
where a better figure is shown.