5. QC Starting from rest, a 5.00-kg block slides 2.50 m down a rough 30.0° incline. The coefficient of kinetic friction between the block and the incline is ur = 0.436. Determine (a) the work done by the force of gravity, (b) the work done by the friction force between block and incline, and (c) the work done by the normal force. (d) Qualitatively, how would the answers change if a shorter ramp at a steeper angle were used to span the same vertical height? Explanation of Solution Given Info: The mass of the block is 5.00 kg. The length of the surface that the block slides is 2.50 m. The inclination of the surface is 30.0°. L= 2.50 m in »,-»| mgv 30.0° Formula to calculate the work done is, W = (F cos 0) d (I) O is the angle between the force vector and the displacement vector • Fis the force • dis the magnitude of displacement Since, the gravitational force acting on the object is, F = mg • m is the mass of the block • gis acceleration due to gravity Thus, equation (I) gives, W = (mg cos 0) d Substitute 5.00 kg for m, 9.8 m/s² for g, 30.0° for 0 and 2.50 m for d to find the work done, W = (5.00 kg) (9.8 m/s²) (sin 30.0°) (2.50 m) = 64.3 J Thus, the work done is 64.3 J. Conclusion: The total work done by the factor of gravity is 64.3 J.

e-ikx and f (x) I give you a function called Q(£, p) and tell you the rules are: î = x, and p = -iħ-. Simply plug in the function Q and Let us play a game. I give you two functions. Let g(x) = e-ax? follow the standard rules of calculus! Work out the following operations on the specified functions. a) £?g(x) b) p²f(x) c) pg(x) d) pf(x) e) &ôg(x) f) Pâg(x) h) &pƒ(x) i) p&ƒ (x) k) Is it true that the operation of âp on a function is always equal to pâ? Justify your answer.

A.) i. Define the plane polar unit vectors ŕ and 0 and show that they are orthonormal. Determine dr dê and de de ii. Hence show that for the vector r(t) = r(t)ŕ dr · = rî +rðê, d²r and = - dt dt² († − rė²)ŕ + (2ŕė +rë)ê . iii. If a mass m moves in the plane along r(t) = a and (t) = 5t determine the shape of the motion and find its period. B) Show that the motion of an object under the universal gravitational force μη Fg is governed by the radial and angular equations of motion and μ = 0, 2r0+ rö = 0. Setting r(t) =R for some constant R, solve these equations of motion and determine the period of the motion. C) Consider the following non-linear differential equation *+5= x +4. Find the stationary solutions. Expand around each of the stationary so- lutions that you have found, in order to determine if they correspond to stable or unstable equilibria. D) Show that 1 5 E = 2 - 4x is a conserved quantity. Hence draw the phase-space diagram and de- termine for which values of E oscilations…

Use Stokes' theorem to evaluate SF. dr where F = z²î + y²ĵ + xk and C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1). The unit vector normal is upward. Z -3 2 2 X