Matlab: Class-9

1. The population x of a certain city satisfies the law dxdt

= 1

100x−

1

108x2, where t is measured in

years. Given that the population of this city is 100,000 in 1980. Plot the populations vs year1980 ≤ t ≤ 2100. In what year does the 1980 population double.

2. A sky-driver equipped with parachute and other essential equipment falls from rest towardsearth. The total weight of the man plus the equipment is 730 N. Before the parachute opens,the air resistance is numerically equal to 0.5v, where v is the velocity (m/s). The parachuteopens after 5sec after the fall begins; after it opens the air resistance is numerically equal to5

8v2. Plot displacement and velocity of the sky-driver with time (0 ≤ t ≤ 25)

3. An 8 N weight is placed upon the lower end of a coil spring suspended from the celling. Theweight comes to rest in its equilibrium position, there by stretching the spring 6 cm. Theweight is then pulled down 3 cm below its equilibrium position and released at t=0 with aninitial velocity of 1cm/s directed downward. Neglecting the resistance of the medium andassuming no external forces are present. Plot the position as a function of time.

4. An 8 N weight is placed upon the lower end of a coil spring suspended from the celling. Theweight comes to rest in its equilibrium position, there by stretching the spring 6 cm. Theweight is then pulled down 3 cm below its equilibrium position and released at t=0 with aninitial velocity of 1cm/s directed downward. The resistance of the medium is numericallyequal to 4dx

dt. No external forces are present. Plot the position as a function of time.

5. An 8 N weight is placed upon the lower end of a coil spring suspended from the celling. Theweight comes to rest in its equilibrium position, there by stretching the spring 6 cm. Theweight is then pulled down 3 cm below its equilibrium position and released at t=0 with aninitial velocity of 1cm/s directed downward. The resistance of the medium is numericallyequal to 4dx

dt. Beginning at t=0 an external force given by F(t) =5 cos2t is applied to the system.

Plot the position as a function of time.

6. A mechanical vibrational system is governed by differential equation y′′ + 4y = 3cos(2t).Plotthe solution of the differential equation , given y(0)=0 and y’(0)=0; 0 ≤ t ≤ 50. Comment onthe solution.

7. Consider the equation of motion of a pendulum θ̈ + g

Lsin(θ) = 0. Solve for position and

velocity of the pendulum for 0 ≤ t ≤ 10 s. Assume that he pendulum is initially at rest and isalmost horizontal. Plot the position and velocity as a function of time.