Consider holding a carton of milk in your hand. The force of the bicep muscle actsat an angle of 15 degrees to the vertical, while the weight of the arm and the milk both act downwards. The distance from the elbow to where the bicep muscle is attached via the distal bicep tendons to the radius and ulna bones is 5 cm. the distance from the elbow to the hand, holding the milk, is 35 cm. the forearm has a mass of 4 kg and the milk carton a mass of 2 kg. Assuming the forearm is kept perfectly horizontal, find the tension in the bicep muscle. As a function of the angle of the forearm with respect to the horizontal direction ( as the forearm is lowered) calculate the tension in the bicep muscle. Include a plot of tension in the bicep as a function of the angle of the forearm relative to the horizontal
To find the tension in the bicep muscle while holding the carton of milk, we need to analyze the forces acting on the forearm.
First, let's break down the forces involved:
1. The weight of the forearm: This force acts downward and can be calculated as the mass of the forearm (4 kg) multiplied by the acceleration due to gravity (9.8 m/s^2). This gives us a downward force of 39.2 N (Newtons).
2. The weight of the milk carton: Similarly, this force acts downward and can be calculated as the mass of the milk carton (2 kg) multiplied by the acceleration due to gravity (9.8 m/s^2). This gives us a downward force of 19.6 N.
3. The force of the bicep muscle: This force acts at an angle of 15 degrees to the vertical and opposes the combined downward forces of the forearm and milk carton.
To calculate the tension in the bicep muscle, we can consider the vertical and horizontal components of the forces.
Vertical forces:
The vertical component of the bicep muscle force helps counteract the vertical downward forces (forearm weight + milk carton weight). However, since the forearm is kept perfectly horizontal, the vertical forces must balance each other out.
Therefore, the vertical component of the bicep muscle force is equal to the sum of the downward forces: 39.2 N + 19.6 N = 58.8 N (directed upward).
Horizontal forces:
The horizontal component of the bicep muscle force is responsible for balancing the torque caused by the weight of the forearm and the milk carton. The torque is calculated as the product of the perpendicular distance from the pivot point (elbow) and the weight of the object.
For the forearm:
Torque = force × distance
Torque = 39.2 N × 0.05 m (distance from elbow to bicep attachment point)
Torque = 1.96 Nm (Newton meters)
For the milk carton:
Torque = force × distance
Torque = 19.6 N × 0.35 m (distance from elbow to hand)
Torque = 6.86 Nm
To find the total torque, we add the torques caused by the forearm and the milk carton: 1.96 Nm + 6.86 Nm = 8.82 Nm.
The horizontal component of the bicep muscle force must be equal to this total torque.
Now, we can use trigonometry to find the horizontal component of the bicep muscle force.
Tension in the bicep muscle = Horizontal component of the bicep muscle force = Torque / perpendicular distance
Tension in the bicep muscle = 8.82 Nm / 0.05 m
Tension in the bicep muscle = 176.4 N
Therefore, when the forearm is held at an angle of 15 degrees to the vertical, the tension in the bicep muscle is 176.4 N.
To plot the tension in the bicep muscle as a function of the angle of the forearm relative to the horizontal, we can calculate the tension for different angles and create a graph. The range of angles will depend on how far the forearm is lowered.