State how many degrees of freedom the following systems have. In each case indicate appropriate generalizedcoordinates which can be used to describe the motion. (You cannot just say that the generalized coordinatesare "r", "0" etc. You should indicate what they are using diagrams.)(a) A particle constrained to move on a plane.(b) A spring pendulum contained in the (x, y) plane, where a mass is attached to an endpoint of a spring ofelastic constant k and equilibrium length 1. The other endpoint of the spring is fixed at the origin.(c) A (pointlike) satellite moving in the gravitational field of Earth.

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**Problem 3.23 Consider the following Hermitian matrix: 2 i 1 T = -i i 1 -i 2 (a) Calculate det(T) and Tr(T). (b) Find the eigenvalues of T. Check that their sum and product are consistent with (a), in the sense of Equation 3.82. Write down the diagonalized version of T. (c) Find the eigenvectors of T. Within the degenerate sector, construct two linearly independent eigenvectors (it is this step that is always possible for a Hermitian matrix, but not for an arbitrary matrix-contrast Problem 3.18). Orthogonalize them, and check that both are orthogonal to the third. Normalize all three eigenvectors. (d) Construct the unitary matrix S that diagonalizes T, and show explicitly that the similarity transformation using S reduces T to the appropriate diagonal form.
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A block of mass m = 1 kg is sliding at 2 m/s [E] on a frictionless surface. A curved frictionless ramp of unknown mass M is at rest on but not attached to the same frictionless surface, as shown at right. The block slides up the ramp after reaching it. 1) The block rises to a maximum height h = 0.125 m on the ramp. What is the velocity of the block at the instant it is at that maximum vertical height? 2) What is the velocity of the block at the instant the block loses contact with the ramp?
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A frog jumps at an angle 30° above the horizontal. The origin of the coordinate system is at the point where the frog leaves the ground. Complete the attached table by drawing check marks in the cells that correctly connect the quantities in the first column that describe the motion of the frog and the descriptions of what is happening to these quantities while the frog is moving. For each quantity, explain why your choice is correct. Consider the frog as a point-like object and assume that the resistive force exerted by the air is negligible.
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State how many degrees of freedom the following systems have. In each case indicate appropriate generalizedcoordinates which can be used to describe the motion. (You cannot just say that the generalized coordinatesare “x”, “θ” etc. You should indicate what they are using diagrams.)(a) A particle constrained to move on a plane
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Disclaimer: if this problem is chosen for the mini-exam, the given function may be different, but will be chosen from the exponential, trigonometric, or polynomial families. * There are two particles (1 and 2) that are moving around in space. The mass of particle 1 is m1 and the mass of particle 2 is m2. The particles are isolated so that only the forces between the particles are significant. The force that 2 exerts on 1 is given by: 2. F1(t) = Fo e î + sin where F, and t, are both constants. The particles interact with one another from t = 0 until t = t1. a. What is the physical meaning of F,? b. Find the impulse from particle 1 on particle 2 over this time interval. c. Find the impulse from particle 2 on particle 1 over this time interval. d. Find the change in momentum of particle 1 over this time interval. e. Find the change in momentum of particle 2 over this time interval. f. Find the change in momentum of the system that consists of particle 1 and particle 2 over this time…
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1.1 Let's say a cart's position is given by x(t) at3 – Bt. (1) - Assume the constants, a and B, are positive. (a) What must be the general type of unit (like length, mass, time, and combinations of these) for each of the constants, a and B? (b) Draw, by hand, a sketch of the two functions at and –ßt for t > 0. Add the two curves graphically to get a final sketch of x(t). (c) Calculate the cart's velocity vr and acceleration ar. (d) Find an expression for the time t = to at which the cart is momentarily at rest. Check: If a and B had the values 1 and 27, respectively (in the correct SI units), you would find t = 3 s. (e) Give the expression for the cart's position x(to) at the time t = to. Always simplify your answer, - and confirm that the units are correct.