A MAN HOLDS A 178 N BALL IN HIS HAND, WITH THE FOREARM HORIZONTAL. HE CAN SUPPORT HE 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 GRAVITY. FIND THE MAGNITUDE OF M AND B)THE MAGNITUDE AND DIRECTION OF THE FORCE APPLIED BY THE UPPER ARM BONE TO THE FOREARM AT THE ELBOW JOINT.
To solve this problem, we need to use the concept of equilibrium. When an object is in equilibrium, the net force and net torque acting on it are both zero.
Let's start with part A, finding the magnitude of the flexor muscle force, M.
1. Begin by drawing a free-body diagram of the system. Label all the forces acting on the forearm and the ball separately.
a) The weight of the ball is 178 N and acts downwards.
b) The weight of the forearm is 22 N and acts downwards.
c) The flexor muscle force, M, acts perpendicular to the forearm, pointing upwards.
d) The force applied by the upper arm bone to the forearm at the elbow joint is the force we need to find. We'll label it as F.
2. Since the forearm is held horizontally, the forces of weight and the flexor muscle force should balance each other. So, the upward flexor muscle force, M, should equal the total downward force on the forearm.
M = Weight of the ball + Weight of the forearm
= 178 N + 22 N
= 200 N
Therefore, the magnitude of the flexor muscle force, M, is 200 N.
Now, let's move on to part B, finding the magnitude and direction of the force applied by the upper arm bone to the forearm at the elbow joint.
3. Since the forearm is in equilibrium, the net torque acting on it must also be zero. The torque caused by the flexor muscle force, M, is balanced by the torque caused by the weight of the forearm and the weight of the ball.
Torque caused by M = Torque caused by forearm weight + Torque caused by ball weight
The torque equation is given by:
Torque = Force × Perpendicular distance from the pivot point
Let's assume the pivot point is the elbow joint.
The torque caused by M is M × d, where d is the distance from the elbow joint to the line of action of the flexor muscle force (perpendicular distance).
The torque caused by the forearm weight is 22 N × d, where d is the distance from the elbow joint to the center of gravity of the forearm.
The torque caused by the ball weight is 178 N × d, where d is the distance from the elbow joint to the center of gravity of the ball.
Since the forearm is horizontal, the torque caused by the forearm weight and the torque caused by the ball weight will be in opposite directions.
Therefore, we can write the equation as:
M × d = 22 N × d + (-178 N × d)
4. Solve the equation for the value of d.
M × d = 22 N × d - 178 N × d
M × d = -156 N × d
M = -156 N
Since both sides of the equation are equal, we can conclude that the magnitude of the elbow joint force, F, is 156 N.
Note: The negative sign indicates that the force applied by the upper arm bone to the forearm at the elbow joint acts in the opposite direction to the flexor muscle force, M.
Therefore, the magnitude of the force applied by the upper arm bone to the forearm at the elbow joint is 156 N, and its direction is opposite to the flexor muscle force, M.