Please answer the following questions.

Lecture 7 Normal & Tangential Vectors; Central Force Problems Problem 2.7.1 Block on a table A 0.5 kg block moves at a speed of v on a circular path of radius 30 cm on a smooth table because a cord with a mass M cylinder hangs through the centre of the table as shown: a) Draw an FBD for the block b) If the block completes one revolution per second, determine the speed v c) Further determine the mass of the cylinder M Problem 2.7.2 Central Force Vector Form One way to write the position for a block moving in a circle like the previous problem is as cos at sin ct , where @=-is the angular frequency and T is the period of its motion. a) Determine (as functions of time where applicable) the vector velocity and acceleration, unit tangent and normal vectors, and the radius of curvature for a particle moving with position vector r Note: don't substitute in values for @, T, v,r or any other variables; leave them as variables and work out the result generally. It may be helpful to execute the following in maple so the simplifications work out nicer: assume (omega, positive) : assume (r, positive) : interface (showassumed=0) :b) Show (by plotting a trajectory for any appropriate value of T and range of time) that this 1' leads to a circular path of radius r moving counterclockwise (Note: for this part you'll need to specify values of r and T) 3 Lecture 8 Central force motion & Space Mechanics Two objects with masses m1 and m2 and charges C11 and q2 are located at positions r1 and r2. a) Write an expression for the electrostatic force of object #1 on object #2. b) Write an expression for the electrostatic force of object #2 on object #1. c) Write an expression for the gravitational force of object #1 on object #2. d) In FlexPDE, you might write the components rxhat and ryhat of a relative position's unit vector using the following: 1: = sqrt(rx"2+ry"2) rxhat * rx/ (r+0.001) ryhat ry/(r+0.001) In these equations, what is the point, if any, of the +0.00]? Suppose a tiny black hole with a mass of 5e12 kg and charge of +3 mC is initially stationary at position (x,y) = (15 m, 15 m). A small spaceship with a mass of 1500 kg and charge of +5 uC is initially located at the origin and moving with velocity vi = (3, 0) mfs a) Write an expression for the total force of the black hole on the spaceship. b) Why might it be valid to ignore the motion of the black hole in this problem? c) Find the trajectory of the spaceship for the next 30 seconds. Make 3 plots: one of the path it travels (i.e., y vs. x), another of its position, velocity, and acceleration's x-components vs time, and a third of its position, velocity, and acceleration's y-components vs time, each scaled appropriately to see what's happening. d) Is there a point in the path when the electrostatic force becomes stronger than the gravitational one? Explain. Suppose the following elevator has a 60% efcient motor, the elevator cart has a mass of 400 kg, the counterweight has a mass of 200 kg, and the cargo has a mass of 100 kg. ,9: M a) How much electrical power will the motor be consuming while lifting the cart & cargo at a rate of 4 m/s? b) Repeat part a) if the counterweight has a mass of 500 kg instead. c) Why is it not a good idea for this elevator to use a 500 kg counterweight? For the situation in "Elevator Power" with the 200 kg counterweight and 100 kg cargo, a) How much total electrical energy is consumed by this motor when lifting the elevator and cargo 20 m at a speed of4 m/s? b) What happens to the electrical energy used by the motor? Suppose a 20 kg block started from rest at the top of a halfpipe with a radius of 2 m, then slides down the half pipe and comes to rest at an angle of 49 = 30 : a) How much energy did the block lose to friction? b) What must the coefcient of static friction be for the block to be able to hold still at this position on the ramp? A 3 kg collar is connected around a smooth (frictionless) cylinder and attached to a stretched out spring whose unstretched length is l m. k = 200 N/m I515 m a) If the collar is released from rest at point A, determine its speed when it reaches point B. b) If we repeat this experiment with a 10 kg collar, will it reach B with a different speed? Explain. Returning to problem 2.8.2., "Central force in FlexPDE", also make expressions for the potential and kinetic energy of the ship, and plot them and their sum vs. time. Hint: The potential energy of a system of two charges separated by a distance r is U = k, 4142 By comparing the coulomb force expression to Newton's law of gravitation, you can work out what the potential energy for a system of two masses must be. Lecture 11 Particle Kinetics - Impulse & Momentum Problem 3.11.1 Toboggan A 40 kg frictionless concrete toboggan starts from rest at point A carrying Gail (60 kg) and Barry (70 kg). 3 m When the toboggan reaches point B, Barry aggressively falls off at a speed of 3 m/s, relative to the toboggan. Find the speed of the toboggan afterwards. Lecture 12 Impact Problem 3.12.1 In the setup below, ball A and ball B each have the same mass. If ball A is released from an angle of $ = 90 Determine the maximum angle ball B will swing to if the coefficient of restitution is e (i.e,. determine it for any coefficient, not for a coefficient of Euler's number).L A B