PHYSICS 1500 FS 2013 Homework #8 (due Wednesday, October 23, 2013 at the beginning of class) 1. Forces and Potential energy: a. Consider a spring with a force represented by F (ky py 3 qy 4 ) ˆj . Is this force conservative? Explain why or why not. Determine the potential energy function if possible. b. Find the force (vector!) associated with the potential energy function U(x,y,z) = 4y2x +3x2+2xy. c. In the fashion of Sec. 8-6, especially Fig. 8-9, discuss the following potential: U(x)= –k2/x + L2/x2, where k and L are constants. Plot the potential and the force as a function of the distance x qualitatively, find the points where they are maximal and minimal, and discuss possible turning points. What is the difference in motion of particles with positive versus negative mechanical energy? If it helps, choose both constants to be one (units?). 2. A block of mass 1kg is sliding down a track consisting of a quarter of a circle of radius 2m which the block slides down on its concave side, followed by a straight horizontal section. The track is set up as follows. The quadrant of the circle is frictionless. The horizontal track has a coefficient of friction of 0.25 up to a point at which the C which is the end of a relaxed, compressible spring of stiffness K. The track under the spring is frictionless. a. Draw the physical situation. b. Assuming the block starts from rest, determine the velocity of the block at the end of the circular part of the track. c. Find the thermal energy produced as the block slides to C. d. What is the velocity of the block just before it hits the spring? e. Calculate the stiffness of the spring. 3. In the test phase of the now-retired space shuttle, a glider of mass 980 kg was used. The glider did glide back to Earth after been released horizontally at an altitude of 3.5km at a speed of 480km/h. The touchdown speed was measured to be 210 km/h. a. Determine the touchdown speed if there hadn’t been any air resistance. b. Find the average force of air resistance assuming the angle of decent was a constant 12 degrees with respect to the surface. (Over) 4. Consider a ½ kg mass glued to the right side of a 60cm long spring of stiffness 40N/m attached to a wall on the left. Initially the mass is at rest at the equilibrium point of the spring on a frictionless surface. When a constant horizontal force is applied to the block, it moves to the right along the surface. a. Draw a diagram of the physical situation and the free-body diagram of the mass. b. Draw the energy bar chart and dind the velocity of the block after it has moved 25cm to the right. c. Once it reaches the 25cm mark, the constant force is removed. Draw another energy bar chart to find out how close will it subsequently get to the wall to which the spring is attached. 5. Find the center of mass of a system of 8 particles located at the corners of an imaginary cube placed in the first octant of a three-dimensional coordinate frame with one corner at the origin. Assume the cube’s side length is 2m, and the masses of the particles are determined by squaring the sum of their coordinates in kilograms and adding 1 kg, e.g. the particle farthest from the origin at (2m,2m,2m) )will have a mass (2+2+2)2 kg + 1kg= 37kg. 6. Consider a bullet of mass 0.1kg and a speed of 600 m/s which penetrates a distance 0.15m into a block of wood that is mounted firmly to the ground. Assuming the force stopping the bullet is constant, determine the following: a. Acceleration of the bullet b. Force on the bullet c. Time it takes the bullet to stop d. Impulse of the force e. Ratio between the impulse and the initial momentum of the bullet 7. Anatomy of a car crash: Mr. Somebody’s truck (2300kg), stopped at a red light, was hit from behind by a 930kg car. The bumpers locked and so were the brakes, as the truck and the car skidded 2.8m before coming to a final stop. Estimating the friction coefficient between tires and road to be 0.8, what was the speed of the offender at impact? 8. A radioactive nucleus makes a radioactive decay into a daughter nucleus, a neutrino and an electron. Assume the latter two are emitted at right angles and have momenta of 6.2 x 10-23 kg m/s and 9.6 x 10-23 kg m/s, respectively. What is the direction and magnitude of the momentum of the recoiling daughter nucleus? 9. Consider a projectile of mass 20kg that is fired with initial velocity of 250m/s at an angle of 60 degrees above the horizontal. As it reaches its maximal altitude, it explodes into two equal-mass parts. One of those falls to the ground vertically with vanishing initial speed. a. Calculate the range of the other part of projectile. b. Determine the amount of energy released during the explosion.