PH1011 Tut 4: Forces, Momentum and Impulse 1) [ A J86/II/8] When a body moves through a fluid, a retarding force due to turbulence may be experienced. In the case of a sphere of radius r moving with speed v through a stationary fluid of density ρ which is at rest, this force is F = kρ r! v! where k is a constant. (Note that this formula differs from Stoke's law in the lecture notes). When spherical raindrops fall through still air, all but the smallest experience a retarding force given by the equation above. It is found that drops of a given radius approach the ground with an approximately constant speed, which is independent of the cloud in which they are form. (a) Find an expression for this terminal speed v! in terms of the constant k, the radius r of the drop, its density ρ!, the density of air ρ! of the air and the acceleration of free fall g. (You may neglect the upthrust due to air). Sketch the velocity-time graph of the object from the moment it is released into the fluid. (b) The terminal speed of the raindrop of radius 1 mm is approximately 7 ms!!. In freak storms, hailstones with radii as large as 20 mm may fall. Estimate the speed with which such stones strike the ground. [Take the density of water as 1 10! kg m!! and the density of ice as 9 10! kg m!!. 1
2) [Activity] a) The density of ice is 0.917 g/cm 3, whereas that of seawater is 1.025 g/cm 3. What percent of an iceberg is above the surface of the water? (Discussion) If some ice and seawater are placed in a measuring cylinder, will the seawater level rise, drop or stay the same when all the ice melts? b) When a crown of mass 14.7 kg is submerged in water, an accurate scale reads only 13.4 kg. Is the crown made of gold? c) A spherical iron ball bearing of diameter 4 mm is placed on a piece of Styrofoam floating in a cylinder of oil as shown in the figure below. You may assume that the weight of the Styrofoam is negligible. Density of iron is 7.86 g cm -3 ; density of oil is 0.83 g cm -3. i) Calculate the upthrust acting on the piece of Styrofoam. ii) What is the volume of oil displaced by the piece of Styrofoam and the iron ball bearing? iii) The ball bearing rolls off the piece of Styrofoam and falls into the oil. Will the oil level be higher, lower or the same as before? Explain your answer. Iron ball bearing Original oil level Styrofoam Oil 2
3) [S9.15] The magnitude of the net force exerted in the x direction on a 2.50-kg particle varies in time as shown in the figure below. Find (a) the impulse of the force over the 5.00-s time interval, (b) the final velocity the particle attains if it is originally at rest, (c) its final velocity if its original velocity is 2.00! m/s, and (d) the average force exerted on the particle for the time interval between 0 and 5.00 s. 4) [G9.32] (a) Calculate the impulse experienced when a 65-kg person lands on firm ground after jumping from a height of 3.0 m. (b) Estimate the average force exerted on the person s feet by the ground if the landing is stiff-legged, and again (c) with bent legs. With stiff legs, assume the body moves 1.0 cm during impact, and when the legs are bent, about 50 cm. [Hint: The average net force on her which is related to impulse, is the vector sum of gravity and the force exerted by the ground.] 5) A neutron collides elastically with a helium nucleus (at rest initially) whose mass is four times! that of the neutron. The helium nucleus is observed to move off at an angle θ!" = 40.!!! Determine the angle of the neutron, θ and the speeds of the two particles, v and v!", after the collision. The neutron s initial speed is 6.2 10! m/s. 6) [G9.59] A neon atom (m=20.0 u) makes a perfectly elastic collision with another atom at rest. After the impact, the neon atom travels away at a 55.6 angle from its original direction and the unknown atom travels away at a 50.0 angle. What is the mass (in u) of the unknown atom? [Hint: You could use the law of sines.] 7) [G9.103] A 0.25-kg skeet (clay target) is fired at an angle of 28 to the ground with a speed of 25 m/s. When it reaches the maximum height, h, it is hit from below by a 15-g pellet travelling vertically upwards at a speed of 230 m/s. The pellet is embedded in the skeet. (a) How much higher h does the skeet go up? (b) How much extra distance Δx does the skeet travel because of the collision? Answers to selected questions 1. (a) v! =!!!!"#!!!! (b) 30 m/s 2. (c)(i) 2.58 10!! N (ii) 0.317 cm! (iii) Lower 3. (a) I = 12.0 Ns ı; (b) v! = 4.80!! ı; (c) v! = 2.80!! ı; (d) F!"# = 2.40 N ı 4. (a) (magnitude) 499 Ns, (b) 301 mg = 191737 N, (c) 7 mg = 4459 N! 5. v!" = 1.899 10! m/s; v! = 4.899 10! m/s; θ! = 85.6 6. 39.9 u 7. (a) 8.6 m; (b) 38.4 m 3