A man holds a 194-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 24.7 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.
65
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To solve this problem, we will use the concept of the equilibrium of forces. The condition for equilibrium is that the sum of all forces acting on an object should be zero.
Let's start by considering the forces acting on the forearm and ball system.
(a) The magnitude of the flexor muscle force: Since the forearm and ball are held in a horizontal position, the flexor muscle force will balance the weight of the ball. So,
Flexor muscle force = Weight of the ball = 194 N
(b) The magnitude of the force applied by the upper arm bone to the forearm at the elbow joint: This force is required to balance the weight of the forearm and the ball. In other words,
Force applied by upper arm bone = Weight of the forearm + Weight of the ball
Given that the weight of the forearm is 24.7 N, we can calculate this force:
Force applied by upper arm bone = 24.7 N + 194 N = 218.7 N
So, the magnitude of the force applied by the upper arm bone to the forearm at the elbow joint is 218.7 N.
(c) The direction of the force applied by the upper arm bone: According to the problem description, the forearm is horizontal, which means the force applied by the upper arm bone is perpendicular to the forearm. Therefore, the force applied by the upper arm bone is vertical, with a positive angle counterclockwise from the horizontal.
Hence, the direction of the force applied by the upper arm bone is directly upward, making a positive angle counterclockwise from the horizontal.
To solve this problem, we'll need to consider the forces acting on the system.
(a) To find the magnitude of the flexor muscle force (𝐹𝑚), we need to calculate the net force acting vertically. The net force is the difference between the weight of the ball and the weight of the forearm:
𝐹𝑛𝑒𝑡 = 𝐹𝑏𝑎𝑙𝑙 − 𝑤𝑓
Given that 𝐹𝑏𝑎𝑙𝑙 (weight of the ball) is 194 N and 𝑤𝑓 (weight of the forearm) is 24.7 N, we have:
𝐹𝑛𝑒𝑡 = 194 N − 24.7 N
𝐹𝑛𝑒𝑡 = 169.3 N
Therefore, the magnitude of the flexor muscle force (𝐹𝑚) is 169.3 N.
(b) To find the magnitude of the force applied by the upper arm bone to the forearm at the elbow joint (𝐹𝑢), we need to consider the equilibrium of torques around the center of gravity of the forearm.
Since the forearm is horizontal, the flexor muscle force (𝐹𝑚) and the force applied by the upper arm bone (𝐹𝑢) form a vertical force couple. In this case, the torque due to the flexor muscle force will balance the torque due to the force applied by the upper arm bone.
The torque due to the flexor muscle force (𝑡𝑜𝑟𝑞𝑢𝑒_𝑚) can be calculated as the product of the flexor muscle force (𝐹𝑚) and the distance from the center of gravity of the forearm to the elbow joint (𝑑):
𝑡𝑜𝑟𝑞𝑢𝑒_𝑚 = 𝐹𝑚 × 𝑑
To maintain equilibrium, the torque due to the force applied by the upper arm bone (𝑡𝑜𝑟𝑞𝑢𝑒_𝑢) should be equal in magnitude but opposite in direction to the torque due to the flexor muscle force:
𝑡𝑜𝑟𝑞𝑢𝑒_𝑢 = −𝑡𝑜𝑟𝑞𝑢𝑒_𝑚
The magnitude of the torque due to the force applied by the upper arm bone (𝑡𝑜𝑟𝑞𝑢𝑒_𝑢) can be calculated by rearranging the equation:
𝑡𝑜𝑟𝑞𝑢𝑒_𝑢 = 𝐹𝑚 × 𝑑
Since the magnitudes of the torques are equal:
𝐹𝑚 × 𝑑 = 𝑡𝑜𝑟𝑞𝑢𝑒_𝑢
Given that 𝑑 is the distance from the center of gravity of the forearm to the elbow joint and 𝑑 has a value of 4.39 cm, we can solve for the magnitude of the force applied by the upper arm bone:
𝐹𝑢 = 𝑡𝑜𝑟𝑞𝑢𝑒_𝑢 / 𝑑
Substituting the given values:
𝐹𝑢 = (169.3 N × 4.39 cm) / 4.39 cm
𝐹𝑢 = 169.3 N
The magnitude of the force applied by the upper arm bone to the forearm at the elbow joint is 169.3 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. Since the forearm is horizontal and the force applied by the upper arm bone is vertical, the angle is 90 degrees counterclockwise from the horizontal.
Therefore, the direction of the force applied by the upper arm bone is a positive angle of 90 degrees counterclockwise from the horizontal.