1 Duration: 10 mins NAME: PHY 121 - Quiz 3 Q1: One problem for human living in outer space is that they are apparently weightless. One way around this problem is to design a space station that spins about its center at a constant rate. This creates “artificial gravity" at the outside rim of the station (3 points). A. If the circumference of the space station is 1884 m, how many revolution per minute are needed in order for the “artificial gravity" acceleration to be 9:8 m=sec2. SOLUTION: The circumference can be used to obtain the radius of the space station 2πR = 1884 m R = 300 m (1) The acceleration a for a rotational motion in terms of angular velocity ! 2 v 2 a = g(in this case) = = r! ! = pg=r = 0:1807 radians=sec (2) r The number of revolution required is the frequency of 2π rotation in a minute N = ! ∗ 60sec=(2π) = 0:1807 ∗ 60=(6:28) = 1:726 rev=min (3) B. If the space station is a waiting area for travelers going to Mars, it might be desirable to simulate the acceleration 2 due to gravity on the Martian Surface amars = 3:7 m=sec . Obtain the angular velocity needed in this case. SOLUTION: 2 We adopt the same procedure as we did for Part A except that the acceleration in now amars = 3:7 m=sec . ! = pg=r = 0:111 radians=sec (4) N = ! ∗ 60sec=(2π) = 0:1807 ∗ 60=(6:28) = 1:06 rev=min (5) 2 Q2: A car of mass M traveling at speed v enters a banked turn covered with ice. The road is banked at an angle θ, and there is no friction between the road and the car's tires. Figure is shown below.(3 points). FIG. 1: Car on a banked road. A. What is the radius r of the turn (assuming the car continues in uniform circular motion around the turn)? Express the radius in terms of the given quantities and g (Acceleration due to gravity). SOLUTION: The normal reaction can be split in X and Y components as mv2 N sin(θ) = N cos(θ) = mg (6) r Dividing the above equation we get v2 v2 tan(θ) = ! r = (7) rg gtan(θ) B. Now, suppose that the curve is level (θ = 0) and that the ice has melted, so that there is a coefficient of static friction µs between the road and the car's tires. What is µmin, the minimum value of the coefficient of static friction between the tires and the road required to prevent the car from slipping? Assume that the car's speed is still v and that the radius of the curve is given by r. Express µmin in terms of v; r; M; and/or g. SOLUTION: mv2 v2 µminmg = ! µmin = (8) r rg.