Radius (we will set the radius r :-j: St (a) Two points on the equator are given by φ1 and φ2 wit...
question (c), (d), (e), (f) please. Thanks.
1 Consider a cylinder of mass M and radius a rolling down a half-cylinder of radius R as shown in the diagram (a) Construct two equations for the constraints: i rolling without slipping (using the two angles and θ), and ii) staying in contact (using a, R and the distance between the axes of the cylinders r). (b) Construct the Lagrangian of the system in terms of θ1, θ2 and r and two...
The Problem The single pendulunm Consider the single pendulum shown below. There is a bob at the end of the pendulum, of mass For small oscillations in the case of a frictionless pivot and a rod with negligible mass, the motion of the pendulum over time can be described by the second-order linear ODE where θ is the angle between the rod and the vertical, g is gravity, t is the length of the rod and θ dag /dt2 Q1...
B.2. The surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r, 0, 0) by where θ0 is a constant (which you may assume is less than π/2) (a) Sketch the surface Sc (b) Using the expression show that the vector element of area on Sc is given by -T Sin where [41 (c) The vector field a(r) is given in Cartesian coordinates by Show that on Sc and hence that 4 2 (d) The curved...
(1 point) Let Vf =-8xe-r sin(5y) 20e-x. cos(Sy) j. Find the change inf between (0,0) and (1, π/2) in two ways vf . dr, where C is a curve connecting (0,0) and (1.d2). (a) First, find the change by computing the line integral The simplest curve is the line segment joining these points. Parameterize it: with 03t s 1, r(t)- so that Icvf . di- Note that this isn't a very pleasant integral to evaluate by hand (though we could...
3. 25 points A uniform hoop of mass M and radius r has a uniform rod of mass m and length r welded to it as illustrated. a) Given that the rod is at an angle θ from the horizontal, and that the hoop is rotating at a rate ω and rolling on the floor without slipping, what is the total Kinetic Energy, and total Potential Energy (use the height of the center of the hoop as your datum)? Hints:...
→ (1 point) Let Vf-6xe-r sin(5y) +1 5e* cos(Sy) j. Find the change inf between (0,0) and (1, n/2) in two ways. (a) First, find the change by computing the line integral c Vf di, where C is a curve connecting (0,0) and (1, π/2) The simplest curve is the line segment joining these points. Parameterize it: with 0 t 1, K) = dt Note that this isn't a very pleasant integral to evaluate by hand (though we could easily...
Can you do 3 and 6
Determine whether the following assertions are true or false 1. The double integral JJDy2dA, where D is the disk x2 +y2く1, is equal to π/3 2. The iterated integral J^S 4drdy is equal to 3. The center of mass of the triangular lamina that occupies the region D- 10 4. The triple integral of a function f over the solid tetrahedron with vertices (0,0,0), x < 3,0 < y < 3-2) and has a...
2. (30 points) A very long, straight, solid copper cylinder of radius R (>2R) is oriented with its axis along e z-direction. The cylinder carries a current whose current density is j(r), where r is the radial distance from the cylinder axis. The current density, although symmetric about the cylinder axis, is not constant but varies with r according to 31o a) (10) Obtain an expression for the current /(in terms of Jo, r and R) flowing in a circular...
1. Image charges in sphere We have two charges of magnitude +Q seperated by a distance of 2d, see drawing. a) Find a grounded conducting sphere (potential set to zero) with radius R, where R is the minimum radius needed to neutralize the repulsion from the two charges on each other. Hint: Try to reverse engineer the idea of image charges for a sphere which was discussed in the lectures. Place image charges and find an expression for the force....
Problem 2: Given a collection of data { zNJS R" we define 1. The sample mean of the points is given by 2. The sample variance of the points is given by N 2 3. The covariance matrix of the points is given by Suppose that (N) S R is a collection of data points. Using Lagrange Multipliers, show that the unit vector w for which the set (i.N), where wy, has maximum variance is the normalized eigenvector of Cov(ia)...