What is the equilibrium force exerted by the muscle in holding a ball?

In summary, a human is holding a ball of mass 8kg with a forearm of mass 2kg and a distance of 0.35 meters from the elbow. The forearm is perpendicular to the upperarm and the distance between the elbow and the center of mass is 0.15 meters, with the distance between the elbow and the muscle being 0.05 meters. To find the force of the muscle given equilibrium, the sum of torques is set equal to zero, resulting in a force of 608.22 Newtons. However, an additional upward force from the shoulder is needed to ensure equilibrium of the vertical forces, but it does not contribute to the torques. This results in a downward force of 510.
  • #1
pixietree
6
1

Homework Statement


A human holds a ball of mass 8kg at his hand, so that the forearm is perpendicular to the upperarm.
The distance between the elbow and the center of mass of the forearm is 0.15 meters, the distance between the elbow and the muscle is 0.05 meters and the mass of the forearm is 2 kg. The ball is held 0.35 meters from the elbow (as depicted below).
Xyk50Z.png

Find the force of the mustle given equilibrium.

Homework Equations


$$\sum_{i=1}^n \tau_i=0$$
$$F_{m} = ma$$

The Attempt at a Solution


$$r_{mustle}*F_{mustle} - r_{arm}*F_{arm} - r_{ball}*F_{ball}=0$$
$$F_{mustle} = \frac {r_{arm}*F_{arm} + r_{ball}*F_{ball}}{r_{mustle}}$$
Hence, $$F_{mustle} = \frac {r_{arm}*m_{arm}+r_{ball}*m_{ball}}{r_{mustle}} *g $$
Setting the given data yields $$\mathbf F = 608.22\mathbf y N$$
Here is the part I don't understand: If I am right, the forearm does not move and hence does not accelerate.
But $$F-m_{arm}*g-m_{ball}*g = 608.22 - 2*9.81 - 8*9.81 = 510.12 N$$, hence the sum of forces on the formarm is not zero.
 
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  • #2
Hi pixietree! :oldsmile:

That is correct.
There is an additional upward force in the shoulder, that was left out, to ensure equilibrium of the vertical forces.
However, it doesn't contribute to the torques, since the elbow is on the same line as that force.
 
  • #3
I like Serena said:
Hi pixietree! :oldsmile:

That is correct.
There is an additional upward force in the shoulder, that was left out, to ensure equilibrium of the vertical forces.
However, it doesn't contribute to the torques, since the elbow is on the same line as that force.

Shouldn't there be an additional weight of the muscle downwards at a distance of rmuscle ?
 
  • #4
rock.freak667 said:
Shouldn't there be an additional weight of the muscle downwards at a distance of rmuscle ?

We're looking at the free body diagram of only the forearm.
The force from the shoulder is transmitted to the forearm in the elbow, where it rotates.
The muscle itself is part of the upper arm, which will have some upward force on it from the shoulder, and some torque as well, to keep it in equilibrium.
 
  • #5
I like Serena said:
We're looking at the free body diagram of only the forearm.
The force from the shoulder is transmitted to the forearm in the elbow, where it rotates.
The muscle itself is part of the upper arm, which will have some upward force on it from the shoulder, and some torque as well, to keep it in equilibrium.

Ah yes that makes more sense.
 
  • #6
If I understand you correctly rock.freak667, I got the torques right hence there is some downward force of 510.12 Newtons acted at the elbow. But why does the elbow feel this force downwards? I didn't understand your explanation.
 
  • #7
pixietree said:
If I understand you correctly rock.freak667, I got the torques right hence there is some downward force of 510.12 Newtons acted at the elbow. But why does the elbow feel this force downwards? I didn't understand your explanation.

If we look at just the forearm, we have a couple of downward forces on the right (ball and arm weight), and an upward force on the left (muscle).
With just that the arm would tilt down.
To compensate, we need an downward force at the elbow to counteract the muscle, that has to pull extra strong.
Consequently all forces cancel out (which they must to be in equilibrium), and then the canceled out torque ensure the forearm doesn't tilt.

So we have a downward force on the elbow.
And according to "action = - reaction", that means we have an equal and opposite upward force from the upper arm on the elbow, which gets then transmitted through the upper arm to the shoulder.
 
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1. What is equilibrium force?

Equilibrium force is the force that exists when an object is at rest or moving at a constant velocity. It is the result of the balancing of all the forces acting on an object.

2. How do you calculate equilibrium force?

To calculate equilibrium force, you must first identify all the forces acting on an object. Then, use the formula Fnet = ma, where Fnet is the net force acting on the object, m is the mass of the object, and a is the acceleration of the object (which is zero in equilibrium). The resulting value is the equilibrium force.

3. What is the importance of calculating equilibrium force?

Calculating equilibrium force is important because it helps us understand the overall forces acting on an object and how they are balanced. It is also essential in determining the stability of an object and predicting its motion.

4. How does the direction of forces affect equilibrium force?

The direction of forces does not affect equilibrium force. The magnitude of the forces and their balancing effect are what determine the equilibrium force. However, the direction of forces can affect the motion of an object in equilibrium.

5. Can equilibrium force be negative?

Yes, equilibrium force can be negative. This means that there are more forces acting in the opposite direction, resulting in a net force of zero. It is important to consider the signs of forces when calculating equilibrium force to accurately represent the direction of the forces acting on an object.

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