Question

Jane is sitting on a chair with her lower leg at a \(30.0^{\circ}\) angle with respect to the vertical, as shown. You need to develop a computer model of her leg to assist in some medical research. If you assume that her leg can be modeled as two uniform cylinders, one with mass \(M=\) \(20 \mathrm{kg}\) and length \(L=35 \mathrm{cm}\) and one with mass \(\quad m=\) \(10 \mathrm{kg}\) and length \(l=40 \mathrm{cm},\) where is the CM of her leg?

Short answer

The center of mass of Jane's leg is at approximately (3.33, 29.11) cm from the hip joint.

Step-by-step solution

Understand the Problem Statement

We need to find the center of mass (CM) of Jane's leg, which is modeled as two uniform cylinders. The larger cylinder (thigh) has mass \(M = 20\, \text{kg}\) and length \(L = 35\, \text{cm}\), while the smaller cylinder (calf) has mass \(m = 10\, \text{kg}\) and length \(l = 40\, \text{cm}\). The leg is positioned at a \(30.0^{\circ}\) angle from the vertical.

Define the Coordinate System

Place the origin of the coordinate system at the hip joint, which is the top of the thigh cylinder. The thigh extends vertically downward, and the calf extends down at a \(30^{\circ}\) angle from the vertical.

Calculate CM of Each Cylinder

The center of mass of each cylinder is at its midpoint. Therefore, the CM of the thigh (\(\text{CM}_{\text{thigh}}\)) is at \(\frac{L}{2} = \frac{35}{2} = 17.5\, \text{cm}\). The CM of the calf (\(\text{CM}_{\text{calf}}\)) is further along the line at an additional \(\frac{l}{2} = \frac{40}{2} = 20\, \text{cm}\) from the knee.

Determine Coordinates for Each CM

For the thigh, the coordinates are \( (0, 17.5) \), as it extends vertically. For the calf, convert \(20\, \text{cm}\) along the inclined line to vertical \((y)\) and horizontal \((x)\) components. The coordinates are given by \[ x = 20 \cdot \sin(30^{\circ}) = 10 \text{ cm} \]\[ y = L + 20 \cdot \cos(30^{\circ}) = 35 + 20 \cdot \cos(30^{\circ}) = 52.32 \text{ cm}\].

Calculate the Composite CM

The total mass \(M_t = M + m = 20 + 10 = 30\, \text{kg}\). Use the formula for the center of mass,\[ x_{\text{CM}} = \frac{M \cdot 0 + m \cdot 10}{M_t} = \frac{10 \cdot 10}{30} = 3.33\, \text{cm}\]\[ y_{\text{CM}} = \frac{M \cdot 17.5 + m \cdot 52.32}{M_t} = \frac{20 \cdot 17.5 + 10 \cdot 52.32}{30} = 29.11\, \text{cm}. \]

Conclusion

The center of mass of Jane's leg, using the specified model and assuming it is composed of two connected cylinders, is located at coordinates roughly \((3.33, 29.11) \) cm from the origin (hip joint).

Key concepts

Physics

In physics, the concept of the center of mass (CM) is crucial, especially when analyzing complex systems. The center of mass represents the average position of the entire mass of an object or system of objects. Imagine balancing a seesaw; the CM would be where you could place a pivot for perfect equilibrium.

In Jane's leg problem, calculating the CM helps in understanding how forces and movements are distributed across her leg. This is significant for medical research as it affects how the leg moves and supports weight. The CM is determined by considering that all parts of the system (in this case, the two cylinders of the leg) influence the total position based on both their mass and placement.

• It is particularly important in designing prosthetics or in treatments of leg movement disorders.
• Knowing the CM can help predict how the leg will react to various stresses.

Cylindrical Model

A cylindrical model is a simplification used in physics to make complex shapes or systems easier to analyze. Consider Jane's leg. Instead of dealing with irregular shapes, it is modeled as two simple cylinders. Each part of the leg corresponds to a cylinder with uniform density and shape.

This model is used because:

• Cylinders have a straightforward way to calculate their center of mass, which is at their midpoint.
• This model allows for easier mathematical treatment, saving time and reducing the possibility of errors.

While simplistic, using cylindrical models provides a solid approximation for analyzing real-world problems. It allows physicists and engineers to break down complex shapes into manageable pieces and then recombine them to understand the system as a whole.

Vertical and Horizontal Components

When dealing with objects at an angle, it’s crucial to understand how forces and distances break down into vertical and horizontal components. Jane’s calf is at an angle; thus, its center of mass must be calculated with respect to both the vertical and horizontal.

To find these components:

• The vertical component, often noted as the y-component, is calculated using the cosine of the angle since it corresponds with the adjacent side in trigonometry.
• The horizontal component, or x-component, uses the sine of the angle, which aligns with the opposite side.

These calculations help pinpoint the exact location of the center of mass in situations where the object isn’t aligned perfectly with the traditional coordinate axes.

Understanding how to dissect the position into these components is incredibly useful for accurately predicting behavior in non-linear systems like limb movements, making it crucial for fields such as robotics and biomechanics.