Problem 1- The half-life of a radioactive material is the time required for an amount of this material to decay to one-half its original value. Show that for any radioactive material that decays according to the equation dQ/dt = -rt, the half-life ?? and the decay constant ?? satisfy the equation: rτ = ln 2.
Problem 2- A can of soda is taken from the refrigerator, and is left in a room whose temperature is 24°C. After half an hour the temperature of the can was 12°C and after another half an hour the temperature of the can was 16°C. What is the temperature inside the refrigerator?
Problem 3- A cup of coffee with initial temperature 90 °C is placed in a room whose temperature is 20°C. Let τ1 be the amount of time passed from the moment that the temperature of the coffee is 90°C till the moment that the temperature of the coffee is 75°C. Let τ2 be the amount of time passed from the moment that the temperature of the coffee is 75°C till the moment that the temperature of the coffee is 60°C. Which of the following is true? (Argue your answer.) a. τ1 > τ2 b. τ1 = τ2 c. τ1 < τ2
Problem 4- A tank of capacity 2000 liters initially contains 1000 liters of water with 30 kg of salt dissolved in it. A solution with a salt concentration of 0.02 kg/liter enters the tank at a rate of 50 liter/hour. A well-stirred solution leaves the tank at a rate of 25 liter/hour. How much salt is in the tank in the moment that it overflows?
Problem 5- Consider a system of two tanks (A and B) as shown in the figure, where initially the tank A contains 200 liters of brine with 4 kg of salt is dissolved, while tank B initially contains 200 liters of pure water. Pure water starts entering tank A at 1 liter/min and a well-stirred mixture leaves the tank A at the same rate (1 liter/min) entering tank B. Meanwhile well-stirred water leaves tank B at the same rate (1 liter/min).
a. Express the amount of salt in tank A as a function of time.
b. Express the amount of salt in tank B as a function of time.
c. At which moment the amount of salt inside tank B is the largest possible?
Problem 6- According to the Torricelli law, the water in an open tank will flow out through a small hole in the bottom with the velocity it would acquire in falling freely from the water level to the hole (in other words v = √(2gh), where g is the free-fall acceleration ≈ 9.8 m/s2).
1. For of a cylindrical tank with radius 10 cm and initial height of the water 40 cm with a small hole of area 1 cm2 at its bottom:
a. express the height of the water h(t) as a function of time
b. find the time of complete draining for this tank
2. For of half-spherical tank with radius 10 cm which is full in the initial moment with a small hole of area 1 cm2 at its bottom:
a. express the height of the water h(t) as a function of time
b. find the time of complete draining for this tank
Problem 7- A mass of weight 2 kg is attached to a spring whose coefficient is k = 20 N/m and then the entire system is submerged in a liquid that exerts a damping force numerically equal to 8 times the instantaneous velocity. Determine the equation of the of the mass if:
a. the mass is initially released with no initial velocity from a point 0.5 meters below the equilibrium position.
b. the mass is initially released from a point 0.8 meters below the equilibrium position with an initial upward velocity of 4 m/s.
Problem 8-
a. A series circuit contains a resistor R = 40Ω, an inductor L = 1H, a capacitor C = 0.008F, and a 15-V battery. If the initial charge and current are both 0, find the charge and current at time t.
b. Using the same specifications as in the above question, but replacing the battery with a generator providing a voltage of E(t) = 10 cos 8t, find the charge and current at time t.