Classical Mechanics: Motion, force and energy.

1. A flea is able to jump straight up about 0.64 m. It has been said that if a flea were as big as a human, it would be able to jump over a 100 story building! When an animal jumps, it converts work done in contracting muscles into gravitational potential energy (with some steps in between). The maximum force exerted by a muscle is proportional to its cross-sectional area, and the work done by the muscle is this force times the length of contraction. If we magnified a flea by a factor of 850, the cross section of its muscle would increase by 8502 and the length of contraction would increase by 850. How high would this "super flea" be able to jump? (Don't forget that the mass of the "superflea" increases as well.)

2.A projectile is launched with a speed of 35 m/s at an angle of 62° above the horizontal. Use conservation of energy to find the maximum height reached by the projectile during its flight.

3.Two blocks, A and B (with mass 50 kg and 100 kg, respectively), are connected by a string, as shown in Figure P5.64. The pulley is frictionless and of negligible mass. The coefficient of kinetic friction between block A and the incline is µk = 0.29. Determine the change in the kinetic energy of block A as it moves from C to D, a distance of 15 m up the incline if the system starts from rest.

4. Energy is conventionally measured in Calories as well as in joules. One Calorie in nutrition is 1 kilocalorie, which we define in Chapter 11 as 1 kcal = 4,186 J. Metabolizing 1 gram of fat can release 9.00 kcal. A student decides to try to lose weight by exercising. She plans to run up and down the stairs in a football stadium as fast as she can and as many times as necessary. Is this in itself a practical way to lose weight?

To evaluate the program, suppose she runs up a flight of 90 steps, each 0.150 m high, in 65.0 s. For simplicity, ignore the energy she uses in coming down (which is small). Assume that a typical efficiency for human muscles is 20.0%. This means that when your body converts 100 J from metabolizing fat, 20 J goes into doing mechanical work (here, climbing stairs). The remainder goes into internal energy. Assume the student's mass is 66.0 kg.
(a) How many times must she run the flight of stairs to lose 1 pound of fat?
(b) What is her average power output, in watts and in horsepower, as she is running up the stairs?

5. A hummingbird is able to hover because, as the wings move downwards, they exert a downward force on the air. Newton's third law tells us that the air exerts an equal and opposite force (upwards) on the wings. The average of this force must be equal to the weight of the bird when it hovers. If the wings move through a distance of 2.9 cm with each stroke, and the wings beat 67 times per second, determine the work performed by the wings on the air in 1 minute if the mass of the hummingbird is 3.0 grams.

6. A shopper in a supermarket pushes a cart with a force of 27 N directed at an angle of 25° downward from the horizontal. Find the work done by the shopper as she moves down a 44 m length of aisle.