1) Force has dimensions of mass times acceleration

PH301 Homework week10, due 5.12.2012, in workshop1) The Morse Potential is given as:

Show that the Morse Potential has a minimum of U=-U0 at x=r0. [30 marks]

Solution: Do first derivative and set zero to find the condition that x=r0. Calculate second derivative and show that it is positive at x=r0. Also show that U=-U0 at x=r0.2) A block of an unknown material weighs 5.00 N in air and 4.55 N when submerged in water. (a) What is the density of the material? (b) From what material is the block likely to have been made? [25 marks]

Picture the Problem We can use the definition of density and Archimedes principle to find the density of the unknown object. The difference between the weight of the object in air and in water is the buoyant force acting on the object.

(a) Using its definition, express the density of the object:

(1)(2)

Apply Archimedes principle to obtain:

(5)

Solve for :

(2)

Because and :

(4)

Substitute in equation (1) and simplify to obtain:

(3)

Substitute numerical values and evaluate (object:

(4)

(b) From Table 13-1, we see that the density of the unknown material is close to that of lead. (5)

3) Your team is in charge of launching a large helium weather balloon that is spherical in shape, and whose radius is 2.5 m and total mass is 15 kg (balloon plus helium plus equipment). (a) What is the initial upward acceleration of the balloon when it is released from sea level? (b) If the drag force on the balloon is given by, where r is the balloon radius, ( is the density of air, and v the balloons ascension speed, calculate the terminal speed of the ascending balloon. [25 marks]Picture the Problem The forces acting on the balloon are the buoyant force B, its weight mg, and a drag force FD. We can find the initial upward acceleration of the balloon by applying Newtons 2nd law at the instant it is released. We can find the terminal speed of the balloon by recognizing that when

ay = 0, the net force acting on the balloon will be zero.

(a) Apply to the balloon at the instant of its release to obtain:

(1)

(4)

Using Archimedes principle, express the buoyant force B acting on the balloon:

(3)

Substitute in equation (1) to obtain:

Solving for ay yields:

(2)

Substitute numerical values and evaluate ay:

(3)

(b) Apply to the balloon under terminal-speed conditions to obtain:

(4)

Substitute for B:

(4)

Solving for vt yields:

(3)

Substitute numerical values and evaluate v:

(2)

4) A 1.5-kg block of wood floats on water with 68 percent of its volume submerged. A lead block is placed on the wood, fully submerging the wood to a depth where the lead remains entirely out of the water. Find the mass of the lead block. [25 marks]

Picture the Problem Let m and V represent the mass and volume of the block of wood. Because the block is in equilibrium when it is floating, we can apply the condition for translational equilibrium and Archimedes principle to express the dependence of the volume of water it displaces when it is fully submerged on its weight. Well repeat this process for the situation in which the lead block is resting on the wood block with the latter fully submerged. Let the upward direction be the positive y direction.

Apply to floating block: (1)(4)

Use Archimedes principle to relate the density of water to the volume of the block of wood:

(4)

Using the definition of density, express the weight of the block in terms of its density:

(2)

Substitute for B and mg in equation (1) to obtain:

(3)

Solving for (wood yields:

(2)

Use the definition of density to express the volume of the wood:

(2)

Apply to the floating block when the lead block is placed on it:, where B( is the new buoyant force on the block and m( is the combined mass of the wood block and the lead block.

(3)

Use Archimedes principle and the definition of density to obtain:

(2)

Solve for the mass of the lead block to obtain:

Substituting for V and (water yields:

(2)

Substitute numerical values and evaluate mPb:

(1)

5) In normal breathing conditions, approximately 5 percent of each exhaled breath is carbon dioxide. Given this information and neglecting any difference in water-vapor content, estimate the typical difference in mass between an inhaled breath and an exhaled breath. Assume that ones lung capacity is about half a liter and that 20% of the air that is breathed in is oxygen. [25 marks]

Picture the Problem One breath (ones lung capacity) is about half a liter. The only thing that occurs in breathing is that oxygen is exchanged for carbon dioxide. Lets estimate that of the 20% of the air that is breathed in as oxygen, is exchanged for carbon dioxide. Then the mass difference between breaths will be 5% of a breath multiplied by the molar mass difference between oxygen and carbon dioxide and by the number of moles in a breath. Because this is an estimation problem, well use 32 g/mol as an approximation for the molar mass of oxygen and 44 g/mol as an approximation for the molar mass of carbon dioxide.

Express the difference in mass between an inhaled breath and an exhaled breath:

where is the fraction of the air breathed in that is exchanged for carbon dioxide.

(10)

The number of moles per breath is given by:

(5)

Substituting for nbreath yields:

(5)

Substitute numerical values and evaluate (m:

(5)

6) How can you determine if two objects are in thermal equilibrium with each other when putting them into physical contact with each other would have undesirable effects? (For example, if you put a piece of sodium in contact with water there would be a violent chemical reaction.) [10 marks]Determine the Concept Put each in thermal equilibrium with a third body; that is, a thermometer. If each body is in thermal equilibrium with the third, then they are in thermal equilibrium with each other.

7) The length of the column of mercury in a thermometer is 4.00 cm when the thermometer is immersed in ice water at 1 atm of pressure, and 24.0 cm when the thermometer is immersed in boiling water at 1 atm of pressure. Assume that the length of the mercury column varies linearly with temperature. (a) Sketch a graph of the length of the mercury column versus temperature (in degrees Celsius). (b) What is the length of the column at room temperature (22.0C)? (c) If the mercury column is 25.4 cm long when the thermometer is immersed in a chemical solution, what is the temperature of the solution? [30 marks]Picture the Problem We can use the equation of the graph plotted in (a) to (b) find the length of the mercury column at room temperature and (c) the temperature of the solution when the height of the mercury column is 25.4 cm.

(a) A graph of the length of the mercury column versus temperature (in degrees Celsius) is shown to the right. The equation of the line is:

(1)

(b) Evaluate

(c) Solve equation (1) for tC to obtain:

Substitute numerical values and evaluate :

8) A thermistor is a solid-state device widely used in a variety of engineering applications. Its primary characteristic is that its electrical resistance varies greatly with temperature. Its temperature dependence is given approximately by R = R0eB/T, where R is in ohms ((), T is in kelvins, and R0 and B are constants that can be determined by measuring R at calibration points such as the ice point and the steam point. (a) If R = 7360 ( at the ice point and 153 ( at the steam point, find R0 and B. (b) What is the resistance of the thermistor at t = 36.85 C? (c) What is the rate of change of the resistance with temperature (dR/dT) at the ice point and the steam point? (d) At which temperature is the thermistor most sensitive? [20 marks]Picture the Problem We can use the temperature dependence of the resistance of the thermistor and the given data to determine R0 and B. Once we know these quantities, we can use the temperature-dependence equation to find the resistance at any temperature in the calibration range. Differentiation of R with respect to T will allow us to express the rate of change of resistance with temperature at both the ice point and the steam point temperatures.

(a) Express the resistance at the ice point as a function of temperature of the ice point:

(1)

Express the resistance at the steam point as a function of temperature of the steam point:

(2)

Divide equation (1) by equation (2) to obtain:

Solve for B by taking the logarithm of both sides of the equation:

and

Solve equation (1) for R0 and substitute for B:

(b) From (a) we have:

Convert 36.85 (C to kelvins to obtain:

Substitute for T to obtain:

(c) Differentiate R with respect to T to obtain:

Evaluate dR/dT at the ice point:

Evaluate dR/dT at the steam point:

(d) The thermistor is more sensitive (has greater sensitivity) at lower temperatures.

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