Final answer:
The force exerted by the muscle during a soccer kick can be found using the equation for torque, given the moment of inertia, angular acceleration, and the effective lever arm. By calculating the torque and dividing it by the lever arm length, the force is determined to be 1184.21 N.
Step-by-step explanation:
The question asks us to find the force exerted by the muscle of a soccer player during a kicking motion, given the angular acceleration, moment of inertia, and the effective perpendicular lever arm. Using the concept of torque in rotational motion from physics, the force can be calculated with the relationship between torque, moment of inertia, and angular acceleration.
Torque (\(\tau\)) is the product of force (F) and the perpendicular distance from the axis of rotation to the line of action of the force, which is the lever arm (r). The formula for torque in terms of angular acceleration (\(\alpha\)) and moment of inertia (I) is \(\tau = I \times \alpha\).
In this case, the moment of inertia (I) is 0.750 kg\(\cdot\)m², the angular acceleration (\(\alpha\)) is 30.00 rad/s², and the lever arm (r) is 1.90 cm, which needs to be converted to meters by dividing by 100, so r = 0.019 m. To find the required force (F), we first find the torque using the formula \(\tau = I \times \alpha\) and then divide the torque by the lever arm (r).
\(\tau = 0.750 \text{ kg\(\cdot\)m}² \times 30.00 \text{ rad/s}² = 22.5 \text{ N\(\cdot\)m}\)
Force is then calculated by dividing the torque by the lever arm (F = \(\tau / r\)).
\(F = 22.5 \text{ N\(\cdot\)m} / 0.019 \text{ m} = 1184.21 \text{ N}\)
Therefore, the force exerted by the muscle is 1184.21 N.