Operation Guide-Lab 2 Torsional Pendulum Contents 1. Familiarize yourself with the operation of the device, Adjust the device carefully so that it is ready for the measurement Such as: learn how to operate the digital timer, adjust the stage so that the device is placed horizontally(please discuss with your partner why we need to do 2. Determine the torsional constant K of the spiral spring with a plastic cylinder 1)Install the object holder. Adjust the chopping bar, so that it can block the light from the diode to the detector when it moves. Measure the time duration for 5 periods of the torsional pendulum with only the object holder to. So the period: To==to Data table no. 1 to/s 2) Measure the mass of the plastic cylinder with an electric balance(one measurement is enough)and its diameter for 3 times with a vernier caliper. Install the plastic cylinder to the object holder. Measure the time duration for 5 periods of the torsional pendulum with the object plastic cylinder t. So the period: T=-t Data table no. 2: erage D/cm / K t The experimental expression of inertia of the object holder lo: I 4 t The theoretical expression of inertia of the plastic cylinder Ipe t The total moment of inertia of the object holder and the plastic cylinder I K 1=10+/ The expression of the torsional constant K can be derived easily as Substitute the measurement results of the qualities of data table 2 into the above expression of K, get the K value and estimate the Uncertainty. Write down the calculation process 3. Measure the moment of inertia of differently shaped objects: a metal barrel, an iron ball and a long-thin metal bar(Optional). Compare the measured results the theoretically alculated values 1) The metal barrel a) Measure the time duration for 5 periods of the torsional pendulum with the object
10 Operation Guide——Lab 2 Torsional Pendulum Contents 1. Familiarize yourself with the operation of the device. Adjust the device carefully so that it is ready for the measurement. Such as: learn how to operate the digital timer; adjust the stage so that the device is placed horizontally (Please discuss with your partner why we need to do so?). 2. Determine the torsional constant K of the spiral spring with a plastic cylinder. 1) Install the object holder. Adjust the chopping bar, so that it can block the light from the diode to the detector when it moves. Measure the time duration for 5 periods of the torsional pendulum with only the object holder t0. So the period: 0 0 1 5 T t . Data table No. 1: No. 1 2 3 Average t0 / s 2) Measure the mass of the plastic cylinder with an electric balance (one measurement is enough) and its diameter for 3 times with a vernier caliper. Install the plastic cylinder to the object holder. Measure the time duration for 5 periods of the torsional pendulum with the object plastic cylinder t. So the period: 1 5 T t . Data table No. 2: No. 1 2 3 Average D / cm t/ s m / g The experimental expression of inertia of the object holder I0: 2 0 0 2 4 K I T ； The theoretical expression of inertia of the plastic cylinder Ipc: 1 1 2 2 2 8 PC I mr mD ； The total moment of inertia of the object holder and the plastic cylinder I: 2 0 2 4 pc K I I I T ； The expression of the torsional constant K can be derived easily as: K= . Substitute the measurement results of the qualities of data table 2 into the above expression of K, get the K value and estimate the Uncertainty. Write down the calculation process. 3. Measure the moment of inertia of differently shaped objects: a metal barrel, an iron ball and a long-thin metal bar (Optional). Compare the measured results with the theoretically calculated values. 1) The metal barrel： a) Measure the time duration for 5 periods of the torsional pendulum with the object
metal barrel t. So the period: T=-I. The previously measured To can be used b) The theoretical equation for the moment of inertia of a barrel is 1,=m(Di+D2). Calculate the experimentally measured value I and the theoretical value I,. Calculate the percentage difference between ther Data table no. 3: 2 Inner diameter D/cm Outer diameter D2/cm t/s 2) The iron ball: a) Measure the time duration for 5 periods of the torsional pendulum with the object iron ball t. So the period: T=-t, Here, the To should be measured again with the small holder only. b) The theoretical equation for the moment of inertia of a ball is /,==mr". Carry out the similar evaluation as above Data table no 4: No. 1 2 3 Average y s to(5×Tos 3) The long-thin metal bar(Optional) Data table no 5: 2 3 Average cm s l/c 4. Verify the parallel axis theorem(Optional) Please design a lab to verify the parallel axis theorem experimentally Hints: Draw a diagram for T versus mdr+md2. Try to derive the Kvalue from the slope and understand the meaning of cross-section point between the obtained line and the y-axis Data table no 6:
11 metal barrel t. So the period: 1 5 T t . The previously measured T0 can be used again. b) The theoretical equation for the moment of inertia of a barrel is 2 2 1 2 1 ( ) 8 t I m D D . Calculate the experimentally measured value I and the theoretical value It. Calculate the percentage difference between them. Data table No. 3: 2) The iron ball: a) Measure the time duration for 5 periods of the torsional pendulum with the object iron ball t. So the period: 1 5 T t . Here, the T0 should be measured again with the small holder only. b) The theoretical equation for the moment of inertia of a ball is 2 5 2 I mr t . Carry out the similar evaluation as above. Data table No. 4: No. 1 2 3 Average D / cm t/ s t0 (5T0)/ s m / g 3) The long-thin metal bar(Optional): Data table No. 5: No. 1 2 3 Average D / cm t/ s t0 (5T0)/ s m / g l / cm 4. Verify the parallel axis theorem (Optional) Please design a lab to verify the parallel axis theorem experimentally. Hints: Draw a diagram for T 2 versus m1d1 2 +m2d2 2 . Try to derive the K' value from the slope and understand the meaning of cross-section point between the obtained line and the Y-axis. Mass m1 = g、m2 = g. Data table No. 6: No. 1 2 3 Average Inner diameter D1/ cm Outer diameter D2/ cm t / s m / g
Position/cm5.0010.0015.0020.0025.00 t/s 3 Average T/s2 midr+mmd ∧04gcm
12 Position / cm 5.00 10.00 15.00 20.00 25.00 t / s (5T) 1 2 3 Average T 2 /s2 m1d1 2 +m2d2 2 /104 g·cm2
Lab 3 The latent heat of vaporization of liquid nitrogen Goal Review the physical concepts and relationships associated with the flow of heat into and out of materials 2. Determine the latent heat of liquid nitrogen(LN 3. Determine the specific heat of copper in the temperature range between liquid nitrogen (LN) and room temperature water Related topics Heat transfer, Specific heat, Latent heat Introduction When heat energy is added to a substance, its temperature usually rises except when a change of phase occurs(e.g, solid melts into liquid or liquid vaporizes into vapor). The phase change of most material occurs without increase or decrease in its temperature. The thermal energy is phase of a matter is called its latent heat. Conversely, as the temperature drops to a point when the phase changes, the latent heat must be released. The latent heat associated with water vapor in the of the m the weather. It moderates temperature drops at night. And when released through the formation of water droplets drives the winds associated with storms. and even hurricane The historical unit of heat energy, the calorie, was defined as the amount of heat energy needed to raise the temperature of one gram of water by one degree Celsius. The calorie is now defined in terms of the sI unit of energy, the joule, by 1cal=4.184J 2. 1 Specific Heat When heat flows into or out of an object, its temperature changes. The connection between the change in heat energy and the change in temperature is the specific heat. The heat energy AO needed to raise the temperature of a substance is related to its mass according to the formula △Q=m△T (1) Where Ao is the quantity of heat entering the material, m is the mass of the material, c is the specific heat, and AT is the change in the temperature of the material. The specific heat c is defined as the heat energy needed to raise the temperature of one gram of a substance by one degree Celsius (C). In general, the value of the specific heat of a solid substance is predominantly a function of temperature, though small variation of the specific heat occurs due to variation in pressure or volume. Usually, the value of c in Eq. (1)is taken as its average value over the temperature interval between its initial and final temperatures, Ti and T respectively
13 Lab 3 The latent heat of vaporization of liquid nitrogen Goal 1. Review the physical concepts and relationships associated with the flow of heat into and out of materials. 2. Determine the latent heat of liquid nitrogen (LN). 3. Determine the specific heat of copper in the temperature range between liquid nitrogen (LN) and room temperature water. Related topics Heat transfer, Specific heat, Latent heat Introduction When heat energy is added to a substance, its temperature usually rises except when a change of phase occurs (e.g., solid melts into liquid or liquid vaporizes into vapor). The phase change of most material occurs without increase or decrease in its temperature. The thermal energy is absorbed to change the state of the matter involved. The amount of energy required to change the phase of a matter is called its latent heat. Conversely, as the temperature drops to a point when the phase changes, the latent heat must be released. The latent heat associated with water vapor in the atmosphere is one of the most significant factors determining the weather. It moderates temperature drops at night. And when released through the formation of water droplets drives the winds associated with storms, and even hurricane. The historical unit of heat energy, the calorie, was defined as the amount of heat energy needed to raise the temperature of one gram of water by one degree Celsius. The calorie is now defined in terms of the SI unit of energy, the joule，by: 1 cal = 4.184 J 2.1 Specific Heat When heat flows into or out of an object, its temperature changes. The connection between the change in heat energy and the change in temperature is the specific heat. The heat energy Q needed to raise the temperature of a substance is related to its mass according to the formula： Q mc T (1) Where Q is the quantity of heat entering the material, m is the mass of the material, c is the specific heat, and T is the change in the temperature of the material. The specific heat c is defined as the heat energy needed to raise the temperature of one gram of a substance by one degree Celsius (℃). In general, the value of the specific heat of a solid substance is predominantly a function of temperature, though small variation of the specific heat occurs due to variation in pressure or volume. Usually, the value of c in Eq. (1) is taken as its average value over the temperature interval between its initial and final temperatures, Ti and Tf respectively
The specific heat of substances varies with the temperature, as for example the average value of the specific heat of aluminum is 0. 17 cal/g.C between room temperature and liquid nitrogen temperature while it remains essentially constant(0.215cal/goC)from room temperature to 100oC In contrast to this, the specific heat of water decreases from 1.00728 cal/g"C to 0.99795 cal/g C in the temperature range 0 to 35.C and then increases to 1.00697 cal /goC at 100oC 2.2 Latent Heat Under certain circumstances the heat supplied to(or removed from) a substance does not cause a change in its temperature, instead it causes a change of phase(e.g, boiling, melting freezing or condensation). The relationship between the heat added and the amount of material that changes from one phase to another is △AO=L (2) Q is the quantity of heat supplied to(or removed from) the material. b. L is the latent heat associated with transformation in question, e. g, Lf(the latent heat of freezing), Ly(the latent heat of vaporization) or Ls(the latent heat of sublimation) c. m is the mass of material which undergoes a transformation of state [liquid to its solid state (freezing) or into its vapor state(vaporization) or the transformation of solid directly into vapor(sublimation) 2.3 Rate of Heat flow If we apply a temperature gradient to an object, one end is kept hot and the other end cold Then heat flows from hot to cold. At a steady state, the rate r(in calories/second or J/s)at which heat flows through the object is given by R=(-)△T Where k is its thermal conductivity [e. g, for styrofoam, k= 6x10-5 cal/(cm.Cs)], A is the area of the material through which the heat transfer takes place, 4x is its thickness, and AT is the temperature difference(Thot -Tcold) 2.4 Determination of the Latent Heat of liquid Nitrogen In this part of the experiment, the value of the latent heat of vaporization L, of liquid nitrogen (Reference value: 47. 8 cal /g) will be determined This part of the experiment will involve the immersion of a copper cylinder at room temperature into liquid nitrogen and measuring the quantity of the liquid nitrogen 4m evaporated as the copper cylinder cools to the temperature of liquid nitrogen (-195.8C). The amount of the heat energy transferred to liquid nitrogen by the copper cylinder should be equal to the heat energy required to evaporate Am grams of the liquid nitrogen. That is
14 The specific heat of substances varies with the temperature, as for example the average value of the specific heat of aluminum is 0.17 cal/g•℃ between room temperature and liquid nitrogen temperature while it remains essentially constant (0.215cal/g•℃) from room temperature to 100℃. In contrast to this, the specific heat of water decreases from 1.00728 cal/g•℃ to 0.99795 cal/g•℃ in the temperature range 0 to 35℃ and then increases to 1.00697 cal／g•℃ at 100℃. 2.2 Latent Heat Under certain circumstances the heat supplied to (or removed from) a substance does not cause a change in its temperature, instead it causes a change of phase (e.g., boiling, melting, freezing or condensation). The relationship between the heat added and the amount of material that changes from one phase to another is, Q L m (2) Where: a. Q is the quantity of heat supplied to (or removed from) the material. b. L is the latent heat associated with transformation in question, e.g., Lf (the latent heat of freezing), Lv (the latent heat of vaporization) or Ls (the latent heat of sublimation). c. m is the mass of material which undergoes a transformation of state [liquid to its solid state (freezing) or into its vapor state (vaporization); or the transformation of solid directly into vapor (sublimation)]. 2.3 Rate of Heat Flow If we apply a temperature gradient to an object, one end is kept hot and the other end cold. Then heat flows from hot to cold. At a steady state, the rate R (in calories/second or J/s) at which heat flows through the object is given by: (3) Where k is its thermal conductivity [e.g., for styrofoam, k= 6×10-5 cal/(cm•℃•s)], A is the area of the material through which the heat transfer takes place, x is its thickness, and T is the temperature difference (Thot – Tcold). 2.4 Determination of the Latent Heat of Liquid Nitrogen In this part of the experiment, the value of the latent heat of vaporization Lv of liquid nitrogen (Reference value: 47.8 cal／g) will be determined. This part of the experiment will involve the immersion of a copper cylinder at room temperature into liquid nitrogen and measuring the quantity of the liquid nitrogen m evaporated as the copper cylinder cools to the temperature of liquid nitrogen (-195.8℃). The amount of the heat energy transferred to liquid nitrogen by the copper cylinder should be equal to the heat energy required to evaporate m grams of the liquid nitrogen. That is: ( ) kA R T x