Question

A crustacean (Hemisquilla ensigera) rotates its anterior limb to strike a mollusk, intending to break it open. The limb reaches an angular velocity of 175 rad/s in \(1.50 \mathrm{ms} .\) We can approximate the limb as a thin rod rotating about an axis perpendicular to one end (the joint where the limb attaches to the crustacean). (a) If the mass of the limb is \(28.0 \mathrm{g}\) and the length is \(3.80 \mathrm{cm}\) what is the rotational inertia of the limb about that axis? (b) If the ex-tensor muscle is 3.00 mm from the joint and acts perpendicular to the limb, what is the muscular force required to achieve the blow?

Short answer

(b) What is the muscular force required to achieve the blow? (a) Based on the solution above, calculate the rotational inertia using the formula \(I = \frac{1}{3}mL^2\) with the given mass and length values. Plug in the values and find the rotational inertia. (b) By following the solution provided, first calculate the angular acceleration, torque, and finally muscular force using the specified formulas. Use the calculated values to determine the muscular force required to achieve the blow.

Step-by-step solution

(a) Calculate the rotational inertia of the limb

To find the rotational inertia of the limb, we need to use the formula used for a thin rod rotating about an axis perpendicular to one end. The formula for the rotational inertia for a thin rod is: \(I = \frac{1}{3}mL^2\) where \(I\) = rotational inertia \(m\) = mass of the limb = \(28.0 \mathrm{g} = 28.0 \times 10^{-3} \mathrm{kg}\) \(L\) = length of the limb = \(3.80 \mathrm{cm} = 3.80 \times 10^{-2} \mathrm{m}\) Plug in the values and find the rotational inertia.

(b) Calculate the angular acceleration

To find the angular acceleration, we will use the formula below: \(\alpha = \frac{\omega^2}{2\theta}\) where \(\alpha\) = angular acceleration \(\omega\) = angular velocity = \(175 \,\text{rad/s}\) \(\theta\) = angular displacement Since the limb reaches an angular velocity of \(175 \,\text{rad/s}\) in \(1.50 \,\text{ms}\), the angular displacement is: \(\theta = \omega t = 175 \,\text{rad/s} \times 1.50 \times 10^{-3} \,\text{s}\) Now, find the angular acceleration.

(c) Calculate the torque acting on the limb

To find the torque acting on the limb, we will use the formula below: \(\tau = I \alpha\) where \(\tau\) = torque \(I\) = rotational inertia \(\alpha\) = angular acceleration Find the torque by plugging in the values.

(d) Calculate the muscular force required to achieve the blow

Finally, to find the muscular force, we will use the relationship between torque and force: \(\tau = Fr\) where \(F\) = muscular force \(r\) = distance from the joint = \(3.00 \,\text{mm} = 3.00 \times 10^{-3} \,\text{m}\) Now, find the muscular force by rearranging the formula and substituting the values: \(F = \frac{\tau}{r}\)