Tipler & Mosca, Physics for Scientists and Engineers, 6th edition, Vol. 1 (Red
book). Chapter 6, sections 6-1, 6-2, and 6-4, especially examples 6-5 and 6-12.
Pre-Lab 5 Work and Energy References This lab concerns the relationship between force exerted through a distance—work—and the change in kinetic energy of a particle. In the lab you will study work from the gravity force, which is constant and work from a rubber band which is a variable force similar to a spring or “Hooke’s Law” force. Physics 121: Tipler & Mosca, Physics for Scientists and Engineers, 6th edition, Vol. 1 (Red book). Chapter 6, sections 6-1, 6-2, and 6-4, especially examples 6-5 and 6-12. Physics 114: Walker, Physics, 4th edition, Vol. 1 (Blue book). Chapter 7, sections 7-1 through 7-3, especially example 7-3, and the discussion around Figures 7-7 through 7-11 (work by a spring).
1. Review of concepts In the following exercises, you may assume that the acceleration of gravity g = 9.81 m/s2, and that there is no friction or air resistance, unless otherwise stated.
Question 1 A block of ice of mass m = 1.2 kg is released from rest at the top of an inclined plane that makes an angle θ = 15 with respect to the horizontal. After it slides (with negligible friction) down the plane by a distance d = 0.8 m, how fast is the ice block traveling? Work this out using the work-energy theorem through the following steps:
a. Draw a free body diagram of the ice block as it slides down the plane. Identify the source of each force in each free body diagram. b. From the free body diagram, calculate the magnitude of the net force acting on it. c. Calculate the work from the net force on the block. d. Use the work-energy theorem to obtain the change in kinetic energy of the block, and thereby obtain the speed of the block after it has traveled a distance d .
©2009 Department of Physics, University of Washington
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Pre-Lab 5: Work & Energy
Question 2 A cart of mass mA = 0.75 kg that includes a force sensor is attached to a spring. It is pulled toward the motion sensor, and released. The force sensor records the force from the spring while the cart is moving, and the computer plots this force F ( x) as a function of position x from the motion sensor. At x1 = 23 cm, F1 = 1.6 N and at x2 = 65 cm, F2 = 0.5 N. a. If the force F ( x) is linear, as shown in the graph, what is the work done by the spring between x1 and x2 ? b. What is the speed of the cart at x2 ? c. If the speed of the cart at x1 were not zero, but 12 cm/s, what would the speed be at x2 ?
2. Apparatus This experiment uses the carts, force sensors and DataStudio, which is the same apparatus that you used in earlier labs. You should review the pre-lab information for those experiments if you are unfamiliar with this equipment.
3. “Error propagation”: how uncertainty in data gives uncertainty in a result Here is a very common situation: you have some measurements—a distribution—from which you have calculated a mean and a standard deviation. You would like to use these measurements to calculate another quantity. What would be the mean and standard deviation of that other quantity? For example, say you have a distribution of velocity measurements v1 , v2 ,…, vn and you have calculated v , the mean, and σ v , the standard deviation. Now you would like to know the mean and standard deviation of the kinetic energy, Ekin = 12 mv 2 . One way to work this out would be to take each velocity measurement, square it, multiply it by
1 2
m , and
get a distribution of energy calculations E1 , E2 ,…, En from which you could find the mean
Ekin and standard deviation σ E . While this is a valid way to proceed, it is likely to be tedious. It
may also be the case that you do not have the original data set but just v and σ v , or it may be that σ v represents a different kind of uncertainty than just the standard deviation, such as instrumental resolution.
Usually it is unnecessary to construct a new distribution every time you want to find the uncertainty in some derived quantity. Instead, you can apply some formulas to the uncertainties that you already know. This process of working out the uncertainty in a later result from the uncertainties in earlier results is called “error propagation.” (The word “error” is a synonym for “uncertainty” in this context, but we avoid it in these notes, since in most other contexts, “error” is the same as “mistake”. Mistakes are avoidable, uncertainty is not.)
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Pre-Lab 5: Work & Energy Mathematical derivations of the error propagation formulas are given in a number of texts and online sources. In these notes we give a more descriptive justification of the formulas. Certainly you should learn the math if you plan to take more physics, but more important is to understand the meaning of the formulas and when it is appropriate to use each one.
Multiplication by a constant: y = Ax Let’s start with the simplest case: a constant multiplier. For example, say you have a measurement of the force from a hanging weight in the form W ± σ W which is in newtons, and you want the mass m ± σ m . Since W = mg , you would you would multiply W by A = 1 g where g is the acceleration of gravity. To find the mean, since you get the same result whether you multiply each number in a sum by a constant and then add them up or you add up the numbers and then multiply the sum by the constant, if y = Ax is the transformation, then y = Ax where A is the constant. The standard deviation is a measure of how spread out the distribution is about the mean. If each number in the distribution is multiplied by the constant, then the spread of the data after the transformation will be proportional to the magnitude of the constant. Thus, If y = Ax then σ y = A σ x
This means that to find the standard deviation σ y of the derived quantity y , you only need to multiply the standard deviation σ x of the original quantity x by the absolute value of the multiplicative constant A . (Note: an additive constant (e.g., y = x + B ) changes only the mean; the standard deviation does not change. Since an additive constant shifts each point in a distribution by the same amount, the spread of the distribution is unaffected.)
Sum and difference: z = x ± y In this case, you have two different measurements and uncertainties that you would like to combine by adding or subtracting them. For example, say you have two sets of force measurements F1 and F2 made by two different force sensors, and you would like to know the net force Fnet = F1 + F2 . The mean is intuitive: it is simply the sum of the means of the two measurements. That is, if the transformation is z = x + y then z = x + y . This follows from the linearity property of averages: the average of a sum is equal to the sum of the averages. The standard deviation is not as intuitive. It is not true that the standard deviation of a sum is equal to the sum of the standard deviations! Don’t make that mistake! Remember that the standard deviation is the square root of the variance, and that the variance is the average of the square deviations. So, the linearity property of averages is applied to the variances, or, following the math out, σ z2 = σ x2 + σ y2 . Thus,
If z = x ± y then σ z = σ x2 + σ y2
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Pre-Lab 5: Work & Energy Note: the claim that the variance of the sum is equal to the sum of the variances is only true when the distributions of the individual terms in the sum are independent. In practice this means that any data point in the set of measurements for x does not depend on the measurement of any data point for y . When a measurement of x depends in some way on y , then σ z2 has another term to account for this effect, called the covariance .
Exponent: y = x n The error propagation formulas so far are exact: they do not depend on the relative magnitudes of the mean to the standard deviation. This is because adding and multiplying by a constant are linear operations. But when the transformation equation is not linear, then the error propagation formulas are subject to the restriction that the standard deviation must be much smaller than the mean, or said another way, the fractional standard deviation must be much smaller than 1:
σx
= εx
| x|
1
The formula for the mean and uncertainty under an exponential transformation is based on this restriction. The idea is that the original distribution could be roughly characterized as running from x − σ x to x + σ x , or equivalently, from x (1 − ε x ) to x (1 + ε x ) . Thus, under the transformation, the distribution for y should run from x n (1 − ε x ) n to x n (1 + ε x ) n . This is approximately the case when ε x is much smaller than 1. If we multiply out the binomial
(1 + ε x )
n
, say, for n = 2 , we get
(1 + ε x )
2
= 1 + 2ε x + ε x2 .
But if ε x is small, say 0.1 (10%), then ε x2 is much smaller, say 0.01 (1%), in this example, and can probably be ignored relative to the other terms 1 and 2ε x . The same is true for other exponents n : for small uncertainty only the term that has the lowest power in ε x is important. This term always has the form “ nε x ”. Thus, we expect the data set to run approximately from
x n (1 − nε x ) to x n (1 + nε x ) . This produces the approximate relationship:
If y = x n and ε x =
σx x
1, then ε y ≈ nε x
The following exercises should help solidify the use of these formulas.
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Pre-Lab 5: Work & Energy
Question 3 The speed of a 0.96 kg cart is measured after it rolls 1.00 m down an inclined track. Five readings are taken, and the average and standard deviation of the speed is computed. The results are v = 0.36 m/s and σ v = 0.02 m/s. a. Calculate the fractional standard deviation in the speed, ε v . b. Calculate the kinetic energy of the cart at the end of the trip, K . c. Calculate the fractional standard deviation in the kinetic energy, ε K . d. Calculate the standard deviation in the kinetic energy, σ K .
Question 4
Two force sensors are attached to a cart pointing in opposite directions. Strings draped over pulleys with weights of different masses hanging on them are attached to the force sensors, as shown in the diagram. The cart is held in place and then released and the force on each sensor is recorded as the car accelerates. The experiment is repeated 4 times and the data are recorded below. Trial
Force on 1 (N, to left)
Force on 2 (N, to right)
1
1.225
0.688
2
1.034
0.594
3
1.455
0.406
4
1.117
0.530
5
1.096
0.465
a. Calculate the mean and standard deviation of the force in each sensor. b. Calculate the net force on the cart from the two strings. (Be careful about direction!) c. Calculate the standard deviation of the net force.
DBP (02/02/2010) 5/5
