# Justify if a mass is negligible.

1. Oct 18, 2012

### thatguythere

1. The problem statement, all variables and given/known data
A standard man holds an object of mass m = 3 kg on the palm of the hand with the arm stretched, as shown in the figure below.
(a) Use the torque equilibrium equation to determine the magnitude of the force F that is exerted by the biceps muscle, when a = 32 cm,b = 6 cm,and the angle θ = 75°.Neglect the weight of the lower arm.
(b) Is the assumption of a negligible mass of the lower arm in part (a) justified?

2. Relevant equations

3. The attempt at a solution
Part a) I completed with little difficulty, I wish only some dialogue on the second part. The way I see it is that even if the man were not holding a mass in his hand, the weight of the lower arm itself would still exert some force on the bicep. Therefore, the mass is not in fact negligible. Any thoughts?

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2. Oct 18, 2012

### Nessdude14

The mass of the arm would be negligible if the arm's mass was tiny relative to the weight, thus causing a small (negligible) error in your final calculations. Taking the mean value from some data about arm weight found on the internet, a human forearm and hand weigh about 1.5 kg, 50% of the weight of the object held in the hand. Although this weight is distributed across the length of the lever arm, I'd agree that this isn't negligible.

If you know integral calculus, you could to a pretty accurate calculation of the force from the muscle with the added weight of the arm by making an integral for the torque from the arm. If you assume the forearm has uniform mass, then:
$\displaystyle dm=(m/L)dx$
where the positive x axis is along the arm, L is arm length, and m is mass of arm. Then:
$\displaystyle dF=g\ dm$
$\displaystyle dτ=x\ dF$
then, substitute to get
$\displaystyle dτ=\frac{m·g}{L}·x\ dx$
then integrate
$\displaystyle τ=\frac{m·g}{L}·\int^L_0x\ dx$
this gives:
$\displaystyle τ=\frac{m·g·L}{2}$

This calculation, along with the arm weight data found on the internet, indicates that the arm weight would increase the torque (and thus the force from the bicep) by about 25%.

Last edited: Oct 18, 2012