Rounding a flat curve Example 5.22 (page 160): The sports car is rounding a flat curve with radius R at a constant speed. If the coefficient of static friction between the tire and road s, what is the vmax without sliding? A5.11 A. an outward centrifugal force of magnitude mv2/R. B. an inward centripetal force of magnitude mv2/R. C. the force of the car’s acceleration. D. The static friction force between tires and road. E. none of the above. A car (mass m) moves at a constant speed v around a flat, unbanked curve of radius R. A free-body diagram for the car should include Uniform circular motion in a vertical circle Example 5.24: A passenger on a carnival ferris wheel moves in a vertical circle of radius R with constant speed v. the seat remains upright during the motion. Find expressions for the force the seat exerts on the passenger at the top and the bottom of the circle. Chapter 6 Work and Kinetic Energy Goals for Chapter 6 To understand and calculate work done by a force To study and apply kinetic energy To learn and use the work-energy theorem To calculate work done by a varying force along a curved path To determine the power in a physical situation Work, a force through a distance As in the illustration, pushing in the same direction that the object moves Use the parallel component if the force acts at an angle Example 6.1 Steve exerts a steady force of magnitude 210 N on the stalled car as he pushes it a distance of 18 m. The car also has a flat tire, so to make the car track straight Steve must push at angle of 30 to the direction of motion. How much work does Steve do? () Steve pushes a second stalled car with a steady force F=(160N)i-(40N)j. The displacement of the car is s=(14m)i+(11m)j. How much work does Steve do in this case? 3300J, 1800 J Work: Positive, Negative, or Zero Work: Positive, Negative, or Zero Total Work Two ways to calculate the total work Wtot by the multiple forces acted on the body: Calculate the work done by the individual forces and then add all the quantities of work. Compute the vector sum of the forces (the net force) and then use this vector sum as to find out the work by it Stepwise solution of work done by several forces Example 6.2 (Page 185) A farmer hitches her tractor to a sled ground. The total weight of sled and load is 14700 N. The tractors exerts a constant 5000-N force at angle of 36.9 above the horizontal. There is a 3500 N friction force opposing the sled’s motiton. Find the individual work and the total work. cos 36.9=0.8, sin 36.9=0.6 A tractor driving at a constant speed pulls a sled loaded with firewood. There is friction between the sled and the road. A. positive. B. negative. C. zero. D. not enough information given to decide Q6.4 The total work done on the sled after it has moved a distance d is The work-energy theorem—Figure 6.8 Work done on an object can change its motion and energy. (a) (b) (c) Kinetic energy and the Work-Energy Theorem We can compare the kinetic energy of different bodies Changes in the energy of a moving body under the influence of an applied force change differently depending on the direction of application. Refer to Figure 6.10. How fast?—Example 6.3 Example 6.3 (page 189). If the total work on the sled (Example 6.2) is 10 kJ, what is the final speed. (4.2 m/s) Forces on a hammerhead—Example 6.4 In a pile driver, a steel hammerhead (m= 200kg) is lifted 3.00 m above I-beam. Then dropped, driving the I-beam 7.4 cm farther into the ground. The vertical rails guiding hammerhead exert a 60-N friction. Use the work-energy theorem find the speed of hammerhead just as it hits I-beam and (b) the average force the hammerhead exerts on the I-beam. (7.55 m/s, 79000N) Different objects, different kinetic energies Two iceboats hold a race on a frictionless horizontal lake. The two iceboats have masses m and 2m. Each iceboat has an identical sail, so the wind exerts the same constant force F on each iceboat. The two iceboats start from rest and cross the finish line a distance s away. Which iceboat crosses the finish line with greater kinetic energy. Work and energy with varying forces—Figure 6.16 In case of constant Fx, The stretch of a spring and the force that caused it The force applied to an ideal spring will be proportional to its stretch. The graph of force on the y axis versus stretch on the x axis will yield a slope of k, the spring constant. Stepping on a scale—Example 6.6 Example 6.6. (Page 194) A woman weighing 600 N steps on a bathroom scale containing a stiff spring. In equilibrium the spring is compressed 1.0 cm under her weight. Find the force constant of the spring and the total work done on it during the compression. (3.0 J) A force of 5N is applied to the end of a spring, and it stretches 10 cm. How much farther will it stretch if an additional 2.5N of force are applied? A) 2.5 cm B) 5 cm C) 10 cm D) 15 cm Work-Energy Theorem for Straight –line Motion, Varying Forces Motion with a varying force—Example 6.7 Example 6.7 (Page 196) An air-track glider of mass 0.1 Kg is attached to the end of a horizontal air track by a spring with force constant 20.0 N/m. Initially the spring is unstretched and the glider is moving at 1.50 m/s to the right. Find the maximum distance d that the glider moves to the right (a) if the air track is turned on so that there is no friction, and (b) if the air is turned off so that there is kinetic friction with uk = 0.47. (0.106 m, 0.086 m and -0.132 m) Work-Energy Theorem for Motion along a Curve Motion on a curved path—Example 6.8 Example 6.8 (page 197) A boy in a swing (w), length of chain is R. You push the boy until the chains make an angle 0 with the vertical. To do this, you exert a varying horizontal force F that starts at zero and gradually increases just enough so that the boy and the swing move very slowly and nearly in equilibrium. What is the total work done on the boy by all forces? What is the work done by the tension force and pushing force. (wR(1-cos0 )) Watt about power? Once work is calculated, dividing by the time that passed determines power. Note the popular culture power unit of horsepower (1hp = 746 w). The energy you use may be noted from the meter the electric company probably installed to measure your consumption of energy in kilowatt-hours. Force and power you depend upon—Example 6.10 Example 6.10, page 200. Each of the two jet engines in a Boeing 767 airliner develops a thrust (a forward force on the airplane) of 197000 N. When the airplane is flying at 250 m/s, what horsepower does each engine develop? (66000 hp) An example you might do if the elevator is out Example 6.11 (Page 200). A 50.0 kg marathon runner runs up the stairs to the top of Chicago’s 443-m-tall sears Tower. To lift herself to the top in 15.0 minutes, what must be her average power output in watts? (241 w)