Question

To keep the calculations fairly simple but still reasonable, we model a human leg that is 92.0 cm long (measured from the hip joint) by assuming that the upper leg and the lower leg (which includes the foot) have equal lengths and are uniform. For a 70.0-kg person, the mass of the upper leg is 8.60 kg, while that of the lower leg (including the foot) is 5.25 kg. Find the location of the center of mass of this leg, relative to the hip joint, if it is (a) stretched out horizontally and (b) bent at the knee to form a right angle with the upper leg remaining horizontal.

Short answer

(a) x ≈ 33.1 cm when stretched; (b) (x, y) ≈ (30.8, 7.3) cm when bent.

Step-by-step solution

Understanding the Problem

We need to determine the center of mass for a human leg modeled as two segments: the upper leg and the lower leg, each with equal lengths of 46 cm (since 92.0 cm divided by 2 equals 46.0 cm). The mass of the upper leg is 8.60 kg, and the mass of the lower leg is 5.25 kg. We will first find the center of mass when the leg is stretched out horizontally and then when it is bent at the knee at a right angle.

Calculate Center of Mass when Stretched Horizontally

When the leg is stretched out horizontally, the centers of mass of each segment can be modeled linearly. Thus, the center of mass for the combined segments (upper and lower leg) can be calculated using the formula for the center of mass: \[ x = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]where:- \( m_1 = 8.60 \) kg and \( x_1 = 23.0 \) cm (center of the upper leg segment),- \( m_2 = 5.25 \) kg and \( x_2 = 69.0 \) cm (center of the lower leg segment, which is 23.0 cm + 46.0 cm). Substitute these values into the equation to find \( x \).

Substituting Values (Stretched)

\[ x = \frac{8.60 \times 23.0 + 5.25 \times 69.0}{8.60 + 5.25} \]Evaluate this expression to find the value of \( x \) which is the center of mass of the leg on the horizontal axis.

Calculate Center of Mass when Bent at Right Angle

When the leg is bent at a right angle, the upper leg is horizontal and the lower leg is vertical. The center of mass in each direction (x and y) needs to be considered.- For the x-direction, the center of mass is located at the center of the upper leg: \( x = \frac{m_1 x_1 + m_2 x_2'}{m_1 + m_2} \) where \( x_2' = 46.0 \) cm because the lower leg does not extend horizontally.- For the y-direction, the center is along the length of the lower leg only: \( y = \frac{m_2 y_2}{m_1 + m_2} \) where \( y_2 = 23.0 \) cm (half of the lower leg length).Write these expressions to find \( x \) and \( y \).

Substituting Values (Bent)

For the x-direction calculation:\[ x = \frac{8.60 \times 23.0 + 5.25 \times 46.0}{8.60 + 5.25} \]For the y-direction calculation:\[ y = \frac{5.25 \times 23.0}{8.60 + 5.25} \]Evaluate both to find the center of mass \((x, y)\).

Key concepts

Human Body Mechanics

Understanding the mechanics of the human body is essential, especially when tackling problems related to the center of mass. The center of mass is essentially the average position of all the mass in an object. In human body mechanics, this is a crucial concept because it helps determine how different parts of the body will behave under various forces. The human leg, for example, is made of several segments, each contributing differently to the overall mass distribution. When these segments move, understanding the physics behind their mass distribution plays a significant role in modeling their mechanics. Such knowledge is applied in fields ranging from medicine to sports science, where knowing how to balance the body's center of mass can make a big difference in performance and injury prevention.

Physics Problem Solving

Physics problem solving often involves breaking down complex phenomena into simpler elements. In this exercise, the problem is broken down into two parts: finding the center of mass when the leg is fully extended and when it is bent at a right angle. The key to solving such problems lies in understanding how different forces and masses interact. This involves applying formulas correctly and thinking about the system both in isolation and as part of a whole. For the human leg, modeling each leg segment and calculating their centers of mass separately eases the complexity. Students often benefit from visualizing the scenario or drawing a diagram, which makes the abstract more tangible and provides a clear path through the problem-solving process.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a physical situation. In this case, we model a leg as two uniform rods. By assigning mass and length to each segment, we form a simplified version of reality that helps us calculate the center of mass. Creating this model involves using the formula for the center of mass, which is: \[ x = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \] This formula allows us to find where the combined mass of the two segments is centered. Modeling simplifies real-world situations and enables the application of mathematical analysis to predict physical behavior. Through this modeling, students not only solve a specific problem but also learn how to apply similar methodologies to a variety of physics problems.

Physics Education

Physics education emphasizes the understanding and application of fundamental principles through problems like the center of mass calculation. This exercise provides a practical scenario where students apply theories to solve real-world issues. By going through step-by-step solutions, students can see how theoretical formulas translate into practicing problem-solving skills. They learn not only calculation techniques but also how to approach physical systems analytically. Such exercises help build critical thinking skills and encourage students to link physics concepts to everyday phenomena. Physics education aims to equip students with the tools to explore and understand the underlying patterns in nature, fostering curiosity and a deeper appreciation of the physical world.