Systems and Center of Mass - AP Physics 1

1. How do external forces affect the center of mass?

   They do not affect its position; only its motion. External forces change motion but not the relative mass distribution.

2. What is the effect of internal forces on center of mass?

   Internal forces do not affect the center of mass. Internal forces are between parts of the system.

3. Explain how to find the center of mass of a composite object.

   Divide into simple shapes, find each center, and use weighted average. Standard approach for irregular or complex geometries.

4. Identify the variable for total mass in a center of mass calculation.

   $M = \sum m_i$. Sum of all individual masses in the system.

5. Identify the formula for the $y$-coordinate of the center of mass.

   $y_{cm} = \frac{\sum m_i y_i}{\sum m_i}$. Same formula as x-coordinate but using y-positions.

6. Calculate the $x_{cm}$ for particles at $(2,0)$ and $(8,0)$ with masses 2 kg and 4 kg.

   $x_{cm} = 6$. Calculation: $\frac{2(2) + 4(8)}{2 + 4} = 6$.

7. How does the center of mass change if a system is rotated?

   It remains unchanged. Rotation doesn't change relative mass positions.

8. Calculate the center of mass for a 4 kg and 6 kg mass at $(2,2)$ and $(6,6)$.

   $(x_{cm}, y_{cm}) = (4.8, 4.8)$. Using formula: $\frac{4(2) + 6(6)}{10} = 4.8$ for both coordinates.

9. What is the center of mass of a uniform circular disk?

   At the center of the disk. Circular symmetry places center of mass at geometric center.

10. Find the center of mass of a hollow cylinder.

    At the midpoint of its axis. Cylindrical symmetry puts center at midpoint of central axis.

11. Calculate the center of mass for equal masses at $(0,0)$, $(4,0)$, $(4,4)$.

    $(x_{cm}, y_{cm}) = (2.67, 1.33)$. Three equal masses: $\frac{0 + 4 + 4}{3} = 2.67$, $\frac{0 + 0 + 4}{3} = 1.33$.

12. What is the role of center of mass in collision analysis?

    Simplifies calculations by considering motion at this point. Center of mass motion represents overall system behavior.

13. What is the significance of the center of mass in motion analysis?

    It simplifies the analysis by treating motion as if all mass is at this point. Allows treating complex systems as point masses.

14. What is the formula for the center of mass of a two-particle system?

    $x_{cm} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2}$. Weighted average of positions using mass as weight.

15. What does the center of mass depend on in a system?

    Mass distribution and positions of the masses. These are the only factors determining center of mass location.

16. Find the center of mass of a 3 kg and 5 kg mass placed at (3,0) and (0,4).

    $(x_{cm}, y_{cm}) = (1.875, 2.5)$. Using formula: $x_{cm} = \frac{3(3) + 5(0)}{8} = 1.875$, $y_{cm} = \frac{3(0) + 5(4)}{8} = 2.5$

17. Define center of mass.

    The point where the total mass of a system is considered to be concentrated. Useful for analyzing motion as if all mass is at this single point.

18. Calculate the center of mass for a uniform rod.

    At the midpoint of the rod. Uniform distribution makes center of mass at geometric midpoint.

19. What happens to the center of mass in uniform acceleration?

    It accelerates uniformly. Center of mass follows Newton's second law like a point mass.

20. What happens to the center of mass if a mass moves within a system?

    The center of mass shifts according to the mass movement. Center of mass follows the moving mass proportionally.

21. Find the center of mass of a uniform square sheet.

    At the center of the sheet. Square symmetry places center at geometric center.

22. Find the $x_{cm}$ for equal masses at $(0,0)$ and $(4,0)$.

    $x_{cm} = 2$. Equal masses make center of mass at midpoint: $\frac{0 + 4}{2} = 2$.

23. What is the center of mass for a symmetric solid sphere?

    At the geometric center of the sphere. Spherical symmetry places center at geometric center.

24. Determine the center of mass of a thin rectangular plate.

    At the intersection of the diagonals. Rectangular symmetry puts center at intersection of diagonals.

25. What condition ensures the center of mass is outside the object?

    Non-uniform mass distribution or hollow structure. Examples include rings, hollow objects, or asymmetric shapes.

26. State the formula for the center of mass of a system of particles.

    $\vec{r}_{cm} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$. Vector form for multiple particles using position vectors.

27. What is the effect of friction on the center of mass location?

    Friction does not affect the location of the center of mass. Friction is an external force affecting motion, not position.

28. Identify the formula for the $z$-coordinate of the center of mass.

    $z_{cm} = \frac{\sum m_i z_i}{\sum m_i}$. Extension to three-dimensional systems using z-coordinates.

29. How is the center of mass related to rotational motion?

    It is the axis about which the object rotates. Objects tend to rotate about their center of mass.

30. Calculate the center of mass for masses 3 kg at (0,0) and 7 kg at (10,0).

    $(x_{cm}, y_{cm}) = (7, 0)$. Weighted average: $\frac{3(0) + 7(10)}{3 + 7} = 7$ for x-coordinate.

31. Determine the center of mass for a uniform square plate.

    At the intersection of the diagonals. Symmetry makes center of mass coincide with geometric center.

32. What is the center of mass formula for continuous mass distribution?

    $\vec{r}_{cm} = \frac{1}{M}\int \vec{r} \ dm$. Integral form for objects with continuous mass distribution.

33. Find the center of mass for masses 2 kg at $(1,0)$ and 8 kg at $(9,0)$.

    $(x_{cm}, y_{cm}) = (7.4, 0)$. Calculation: $\frac{2(1) + 8(9)}{2 + 8} = 7.4$.

34. What is the center of mass for a symmetric object?

    It is at the geometric center of the object. Symmetry ensures equal mass distribution around center.

35. Identify the center of mass for a triangle.

    At the centroid, intersection of medians. For triangles, centroid and center of mass coincide when uniform.

36. What is the center of mass for a uniform rod of length $L$?

    $\frac{L}{2}$. For uniform objects, center of mass is at geometric center.

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