(7%) Problem 10: A soccer player extends her lower leg in a kicking motion by exerting a force with the muscle above the knee in the front of her leg. Suppose she produces an angular acceleration of 30.5 rad/s² and her lower leg has a moment of inertia of 0.75 kg-m² .
Angular Momentum
The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
![**Problem 10:**
A soccer player extends her lower leg in a kicking motion by exerting a force with the muscle above the knee in the front of her leg. Suppose she produces an angular acceleration of **30.5 rad/s²** and her lower leg has a moment of inertia of **0.75 kg·m²**.
What is the force, in newtons, exerted by the muscle if its effective perpendicular lever arm is **2.1 cm**?
**F = [Answer box]**
This problem requires you to calculate the muscle force using the given angular acceleration, moment of inertia, and lever arm. The formula to use is:
\[ \tau = I \cdot \alpha \]
Where:
- \(\tau\) is the torque exerted by the force,
- \(I\) is the moment of inertia,
- \(\alpha\) is the angular acceleration.
The torque can also be calculated by:
\[ \tau = F \cdot r \]
Where:
- \(F\) is the force,
- \(r\) is the effective lever arm (converted to meters).
You will need to equate these two expressions for torque and solve for \(F\).
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