Question

in each of problems 1 through 8: a. find a fundamental matrix for the given system of equations. b. find the fundamental matrix φ(t) satisfying φ(0) = i. 4. x = ?−1 −4 1 −1 ? x

Answer 1

The question involves solving physics problems related to particle motion and center of mass while also mentioning a mathematical concept of a fundamental matrix for a system of equations.

Step-by-step explanation:

The student's question appears to be about solving a set of problems related to particle motion and determining the center of mass in a physics context. However, the initial problem statement regarding a fundamental matrix for a system of equations seems to relate to mathematics rather than physics.

For the physics part, typical steps include:

1. Determining the system of interest.
2. Drawing a free-body diagram to identify all external forces.
3. Applying Σ Ti = Ia, the rotational equivalent of Newton's second law.
4. Solving the relevant equations to find unknowns such as the radii of circles of motion, x- and y-coordinates of the center of mass, and the motion of the center of mass.

For the mathematical part about fundamental matrices, you would typically solve a system of differential equations to find a matrix that satisfies certain initial conditions.

Answer 2

To find the fundamental matrix for the given system of equations, use the formula ƒ(t) = e^(At), where A is the coefficient matrix. For this system, the fundamental matrix is | e^(-t) -e^(-t) | | -4e^(-t) e^(-t) |. To find the matrix satisfying ƒ(0) = i, evaluate ƒ(0) and obtain | 1 -1 | | 0 1 |.

Step-by-step explanation:

To find the fundamental matrix for the given system of equations, we will use the formula:

ƒ(t) = e^(At)

where A is the coefficient matrix of the system of equations.

In this case, the coefficient matrix is:

| ? -1 |

| -4 1 |

Therefore, the fundamental matrix, ƒ(t), is:

| e^(?t) -e^(-t) |

| -4e^(-t) e^(-t) |

To find the fundamental matrix, ƒ(t), satisfying ƒ(0) = i, we need to evaluate ƒ(0) using the above formula:

ƒ(0) = e^(A * 0) = e^0 = 1

Therefore, the fundamental matrix, ƒ(t), satisfying ƒ(0) = i, is:

| 1 -1 |

| 0 1 |