(a) If the length of the Achilles tendon increases 0.55 cm when the force exerted on it by the muscle increases from 2200 N to 6000 N, what is the "spring constant" of the tendon?
N/m
(b) How much work is done by the muscle in stretching the tendon 0.55 cm as the force increases from 2200 N to 6000 N?
J
Could any one please explain this to me I don't know how to do it.Thank you.
Spring constant is defined as k = ∆F/∆x, change in force divided by change in spring length. ∆F = 6000-2200=3800N, and ∆x = 0.55 cm (0.0055 m), so k = 3800/0.0055 N/m =6.9•10^5 N/m
The energy stored in a deflected spring is
kΔx²/2 = 6.9•10^5•3•10^-5/2 =10.35 J ; this must equal the work done in changing the spring length. So for the muscle, W = 10.35 J
(a) To find the spring constant of the tendon, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.
The formula for Hooke's Law is:
F = k * Δx
Where:
F is the force exerted on the tendon,
k is the spring constant of the tendon, and
Δx is the change in length of the tendon.
In this case, we are given the force exerted on the tendon (F1 = 2200 N), the change in length (Δx = 0.55 cm), and the new force exerted (F2 = 6000 N).
We can rearrange the formula to solve for the spring constant (k):
k = F1 / Δx
Substituting the given values:
k = 2200 N / 0.55 cm
However, we need to convert the change in length from centimeters to meters, as the unit of the spring constant is N/m. There are 100 centimeters in a meter, so:
k = 2200 N / (0.55 cm * (1 m/100 cm))
Simplifying this expression, we get:
k = 2200 N / (0.0055 m)
Evaluating the expression, we find:
k = 400,000 N/m
Therefore, the spring constant of the tendon is 400,000 N/m.
(b) To calculate the work done by the muscle in stretching the tendon, we can use the formula:
Work = Force * Distance
In this case, the force (F1) is 2200 N and the distance (Δx) is 0.55 cm. However, we need to convert the distance to meters, so the work formula becomes:
Work = 2200 N * (0.55 cm * (1 m/100 cm))
Simplifying this expression, we get:
Work = 2200 N * 0.0055 m
Evaluating the expression, we find:
Work = 12.1 J
Therefore, the work done by the muscle in stretching the tendon is 12.1 Joules.