Problem 1: When you ride a bicycle at constant speed, nearly all the energy you expend goes into the work you do against the drag force of the air. Model a cyclist as having cross-section areaA = 0.45 m2 and, because the human body is not aerodynamically shaped, a drag coeicient of C = 0.9. a) What is the cyclist's power output while riding at a steady u = 7.3 m/s (16mph)? Use the fact that in order to keep the speed constant, the force that moves the cyclist forward, I-1 (the force of static friction between the wheels and the ground created as a result of pedaling)> should be equal in magnitude and opposite in direction to the drag force, dg. Use = 1.2 kg/m3 for the air density. (Answer: P = 95 W.) b) Metabolic power is the rate at which your body "burns" fuel to power your activities. For many activities, your body is roughly 25% efcient at converting the chemical energy of food into mechanical energy. What is the cyclist's metabolic power while cycling at 73 m/s? c) The food calorie is equivalent to 4190]. How many calories does the cyclist bum if he rides over level ground at 7.3 m/s for 1 h? (Answer: 326 cal.) Problem 2: A gardener pushes an m = 12 kg lawnmower whose handle is tilted up 0 : 37 above horizontal, 'Ihe lawnmower's co- efcient of rolling friction is p, = 0.15. How much power does the gardener have to supply to push the lawnmower at a constant speed ofu = 1,2 m/s? Assume his push is parallel to the handle. "'3 E a) Fig, 1 shows the free-body diagram for the lawnmower I? is the force with which the gardener is pushing on the handle, and FIG 1: The scheme f\" Problem 2 f, is the rolling friction force of magnitude f, = ny,. Write down x- and y-components of Newton's second law for the lawnmower (since the speed is constant, both vertical and horizontal components of the acceleration are zero) and show that the absolute Value of force I3 obeys the formula 1" = (ms 9": ' . 9 . 1h 5m ) b) Using general formula for the power, P = IF ~17, show that the power supplied by the gardener in this problem is P : ? (note that i and 5 do not have the same direction'). c) Compute numerical value of P from the parameters given in the description of the problem. Problem 3: A uniform solid bar with mass m and length L rotates with angular velocity w about an axle at one end of the bar. What is the bar's kinetic energy? a) Fig. 2 shows the diagram for this problem, Focus on the small element of the bar, dm, of width dy, picked in the diagram (the horizontal velocity LT of this element is the result of rotation of the bar). Assuming that the mass is distributed uniformly along the bar, what is dm in terms of y, m and L? What is the speed of the element, 11, in terms of its coordinate y and the angular velocity 1? FIG. 2: The scheme for Problem 3 b) The kinetic energy of the element dm is dK = #. Using your expressions from the previous step, show that it can be brought to the form (1K = \"T'zzyzdyl c) Find the total kinetic energy of the bar as K = [$0 dK (this expression states that the total energy ofthe bar is the sum of the energies of all innitesimal elements along its length). (Answer: K = #)