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Chapter 7: Problem 32

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

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

Step by step solution

01

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.
02

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.
03

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.
04

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 \[ x = 20 \cdot \sin(30^{\circ}) = 10 \text{ cm} \]\[ y = L + 20 \cdot \cos(30^{\circ}) = 35 + 20 \cdot \cos(30^{\circ}) = 52.32 \text{ cm}\].
05

Calculate the Composite CM

The total mass \(M_t = M + m = 20 + 10 = 30\, \text{kg}\). Use the formula for the center of mass,\[ x_{\text{CM}} = \frac{M \cdot 0 + m \cdot 10}{M_t} = \frac{10 \cdot 10}{30} = 3.33\, \text{cm}\]\[ y_{\text{CM}} = \frac{M \cdot 17.5 + m \cdot 52.32}{M_t} = \frac{20 \cdot 17.5 + 10 \cdot 52.32}{30} = 29.11\, \text{cm}. \]
06

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).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 crucial to understand how forces and distances break down into vertical and horizontal components. Jane鈥檚 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鈥檛 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.

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Most popular questions from this chapter

A 75-kg man is at rest on ice skates. A 0.20-kg ball is thrown to him. The ball is moving horizontally at \(25 \mathrm{m} / \mathrm{s}\) just before the man catches it. How fast is the man moving just after he catches the ball?
Two African swallows fly toward one another, carrying coconuts. The first swallow is flying north horizontally with a speed of $20 \mathrm{m} / \mathrm{s} .$ The second swallow is flying at the same height as the first and in the opposite direction with a speed of \(15 \mathrm{m} / \mathrm{s}\). The mass of the first swallow is \(0.270 \mathrm{kg}\) and the mass of his coconut is \(0.80 \mathrm{kg} .\) The second swallow's mass is \(0.220 \mathrm{kg}\) and her coconut's mass is 0.70 kg. The swallows collide and lose their coconuts. Immediately after the collision, the \(0.80-\mathrm{kg}\) coconut travels \(10^{\circ}\) west of south with a speed of \(13 \mathrm{m} / \mathrm{s}\) and the 0.70 -kg coconut moves \(30^{\circ}\) east of north with a speed of $14 \mathrm{m} / \mathrm{s} .$ The two birds are tangled up with one another and stop flapping their wings as they travel off together. What is the velocity of the birds immediately after the collision?
For a system of three particles moving along a line, an observer in a laboratory measures the following masses and velocities. What is the velocity of the \(\mathrm{CM}\) of the system? $$\begin{array}{cc}\hline \text { Mass }(\mathrm{kg}) & v_{x}(\mathrm{m} / \mathrm{s}) \\\\\hline 3.0 & +290 \\\5.0 & -120 \\\2.0 & +52 \\\\\hline\end{array}$$
A system consists of three particles with these masses and velocities: mass \(3.0 \mathrm{kg},\) moving north at \(3.0 \mathrm{m} / \mathrm{s}\) mass $4.0 \mathrm{kg},\( moving south at \)5.0 \mathrm{m} / \mathrm{s} ;\( and mass \)7.0 \mathrm{kg}\( moving north at \)2.0 \mathrm{m} / \mathrm{s} .$ What is the total momentum of the system?
Two identical gliders on an air track are held together by a piece of string, compressing a spring between the gliders. While they are moving to the right at a common speed of \(0.50 \mathrm{m} / \mathrm{s},\) someone holds a match under the string and burns it, letting the spring force the gliders apart. One glider is then observed to be moving to the right at $1.30 \mathrm{m} / \mathrm{s} .$ (a) What velocity does the other glider have? (b) Is the total kinetic energy of the two gliders after the collision greater than, less than, or equal to the total kinetic energy before the collision? If greater, where did the extra energy come from? If less, where did the "lost" energy go?
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