Consider holding a carton of milk in your hand, as shown in the image.
The force of the bicep muscle acts at an angle of 15
o
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 the distal
bicep tendons to the radius and ulna bones is 5 cm. The distance from the
elbow to the center of mass of the forearm is 16.5 cm and 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.
a) Assuming the forearm is kept perfectly horizontal, find the tension in the bicep muscle.
b) 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.
c) Include a plot of tension in the bicep as a function of the angle of the forearm relative to the
horizontal (don't forget to label axis)
To calculate the tension in the bicep muscle, we need to analyze the forces acting on the forearm and solve for the net force in the vertical direction.
First, let's understand the forces involved in this scenario. We have:
- The force of the bicep muscle acting at an angle of 15° to the vertical.
- The weight of the arm and the milk, which both act downward.
a) Assuming the forearm is kept perfectly horizontal, the angle between the force of the bicep muscle and the vertical direction is 90°. Therefore, there is no vertical component of the force of the bicep muscle.
To find the tension in the bicep muscle, we only need to consider the weight of the arm and the milk. The net force in the vertical direction is equal to the sum of the weight of the forearm and the weight of the milk.
Weight of the forearm = mass of the forearm x acceleration due to gravity
Weight of the forearm = 4 kg x 9.8 m/s²
Weight of the forearm = 39.2 N
Weight of the milk = mass of the milk x acceleration due to gravity
Weight of the milk = 2 kg x 9.8 m/s²
Weight of the milk = 19.6 N
Net force in the vertical direction = Weight of the forearm + Weight of the milk
Net force in the vertical direction = 39.2 N + 19.6 N
Net force in the vertical direction = 58.8 N
Since the forearm is kept perfectly horizontal, there is no vertical acceleration, meaning the net force in the vertical direction is zero. Therefore, the tension in the bicep muscle is also 58.8 N.
b) As the forearm is lowered, the angle between the force of the bicep muscle and the vertical direction decreases. Let's call this angle θ.
To calculate the tension in the bicep muscle as a function of the angle of the forearm, we need to find the vertical component of the force of the bicep muscle, which acts against the net force in the vertical direction.
Vertical component of the force of the bicep muscle = Force of the bicep muscle * sin(θ)
Net force in the vertical direction = Weight of the forearm + Weight of the milk
Since the vertical component of the force of the bicep muscle acts against the net force in the vertical direction, we have:
Force of the bicep muscle * sin(θ) = Weight of the forearm + Weight of the milk
We can rearrange the equation to solve for the tension in the bicep muscle:
Force of the bicep muscle = (Weight of the forearm + Weight of the milk) / sin(θ)
c) To plot the tension in the bicep muscle as a function of the angle of the forearm relative to the horizontal, you can use the equation derived in part b) and vary the angle θ from 0° to 90°.
Plotting the angle θ on the x-axis and the tension in the bicep muscle on the y-axis would give you the desired plot. Make sure to label the axes accordingly.
To solve this problem, we need to analyze the forces acting on the forearm and apply the principles of torque and equilibrium.
a) To find the tension in the bicep muscle when the forearm is perfectly horizontal, we need to set up the equation for torque. Torque is defined as the force applied multiplied by the distance from the pivot point.
In this case, the force of the bicep muscle is acting at an angle of 15° to the vertical. We can break this force into its vertical and horizontal components.
The vertical component of the force is given by:
F_vertical = F_bicep * cos(15°)
The horizontal component of the force is given by:
F_horizontal = F_bicep * sin(15°)
The torque created by the bicep muscle is given by:
τ_bicep = F_vertical * d_bicep,
where d_bicep is the distance from the elbow to where the bicep muscle is attached (5 cm).
The weight of the forearm and the milk also create a torque, acting in the opposite direction. The torque created by the weight is given by:
τ_weight = (m_forearm + m_milk) * g * d_center,
where m_forearm is the mass of the forearm (4 kg), m_milk is the mass of the milk carton (2 kg), g is the acceleration due to gravity (9.8 m/s^2), and d_center is the distance from the elbow to the center of mass of the forearm (16.5 cm).
In equilibrium, the torques created by the bicep muscle and the weight should balance each other out. Therefore, we can set up the equation:
τ_bicep = τ_weight
Substituting the expressions for τ_bicep and τ_weight, we have:
F_vertical * d_bicep = (m_forearm + m_milk) * g * d_center
Solving for F_vertical, the vertical component of the force of the bicep muscle, we get:
F_bicep = [(m_forearm + m_milk) * g * d_center] / (d_bicep * cos(15°))
b) To calculate the tension in the bicep muscle as a function of the angle of the forearm, we can use the same equation as in part a, but with a general angle θ instead of 15°.
F_bicep = [(m_forearm + m_milk) * g * d_center] / (d_bicep * cos(θ))
c) To create a plot of tension in the bicep muscle as a function of the angle of the forearm, you can vary the value of θ from 0° to 90° and calculate the corresponding tension using the equation from part b. Then, you can plot the angle θ on the x-axis and the tension in the bicep muscle on the y-axis.
Note: Please provide the value of g (acceleration due to gravity) if it is different from the standard value of 9.8 m/s^2.