The goal in this problem is to find the growth of an ice layer as a function of time. Call the thickness of the ice layer L . (a) Derive an equation for dL/dt in terms of L , the temperature T above the ice, and the properties of ice (which can leave in symbolic form instead of substituting the numbers). (b) Solve this differential equation assuming that at t = 0 , you have L = 0 . If you have studied differential equations, you will know a technique for solving equations of this type: manipulate the equation to get dL/dt multiplied by a (very simple) function of L on one side, and integrate both sides with respect to time. Alternatively, you may be able to use your knowledge of the derivatives of various functions to guess the solution, which has a simple dependence on t . (c) Will the water eventually freeze to the bottom of the flask?
The goal in this problem is to find the growth of an ice layer as a function of time. Call the thickness of the ice layer L . (a) Derive an equation for dL/dt in terms of L , the temperature T above the ice, and the properties of ice (which can leave in symbolic form instead of substituting the numbers). (b) Solve this differential equation assuming that at t = 0 , you have L = 0 . If you have studied differential equations, you will know a technique for solving equations of this type: manipulate the equation to get dL/dt multiplied by a (very simple) function of L on one side, and integrate both sides with respect to time. Alternatively, you may be able to use your knowledge of the derivatives of various functions to guess the solution, which has a simple dependence on t . (c) Will the water eventually freeze to the bottom of the flask?
The goal in this problem is to find the growth of an ice layer as a function of time. Call the thickness of the ice layer L. (a) Derive an equation for dL/dt in terms of L , the temperature T above the ice, and the properties of ice (which can leave in symbolic form instead of substituting the numbers). (b) Solve this differential equation assuming that at
t
=
0
, you have
L
=
0
. If you have studied differential equations, you will know a technique for solving equations of this type: manipulate the equation to get dL/dt multiplied by a (very simple) function of L on one side, and integrate both sides with respect to time. Alternatively, you may be able to use your knowledge of the derivatives of various functions to guess the solution, which has a simple dependence on t. (c) Will the water eventually freeze to the bottom of the flask?
Please solve it correctly! It is of advanced physics.
Problem 1:
How long does it take to heat a cup of coffee in a 1000-Watt microwave oven? This means that
energy is used at the rate of 1000 Joules per second. Assume that the coffee starts at a normal
room temperature of 25°C.
Step 1: Guess an answer.
Step 2: Estimate the volume of the coffee in mL and the final temperature that you want to
attain.
Step 3: Assume coffee has the same density and thermal properties of water. Find its heat
capacity (specific heat times mass) in appropriate units.
Step 4: Use the heat capacity and the desired temperature change to find the energy
required.
Step 5: Calculate the time required using the energy and the microwave power. Pay
attention to units and use Power = Energy/time.
Step 5: Is your answer reasonable?
Imagine a pond initially at 0°C on a winter. The atmosphere has a constant temperature of --13.28°C. A very small portion of mass
dm will freeze at time dt. Here, you can assume that the ice that formed in the pond has uniform density. If the pond is 33 m deep,
how long will it take to freeze the whole pond? Assume that 1yr = 365 days.
Express your final answer in YEARS, and in ZERO decimal place.
The thermal conductivity of ice is 1.6 W/mK, density of ice is 920 kg/m³, and its latent heat of fusion is 334x10³ J/kg
Round your answer to 0 decimal places.
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