SP221/2141&4341 Chapters 1 & 2 Homework Due Thursday September 8, 2016
Question 1: An object dropped from rest will, in the absence of air resistance, fall a distance h =
1 gt
2
2 in time t , where g is acceleration due to gravity. You repeatedly drop an object from a height of 2.00
± 0.03 m and measure how long it takes to hit the ground. Over 12 trials, you measure
0.630 s, 0.645 s, 0.650 s, 0.635 s, 0.640 s, 0.620 s, 0.635 s, 0.650 s, 0.625 s, 0.640 s, 0.645 s, and
0.635 s as the time the object is in the air. What would you report as the experimental value for g , including uncertainty and units? Keep two significant digits in the uncertainty, and an equal number of decimal places in the value. (You might find it easier to calculate the standard deviation using
Excel. Type the command =STDEV.P(....) into a cell where the argument of the command is the list of numbers, either entered manually or by scrolling through a column or row.) [10 points]
Question 2: An object moving through a fluid will experience a drag force, F d
. The dimensions of force are given in Table 1-2 of your textbook. [10 points]
(a) If an object is moving at low speed through a material with high viscosity the only parameters that can affect F d are the object’s speed ( v ), its cross-sectional area ( A ), and the fluid’s dynamic viscosity ( µ ). This limit is known as Stokes’ drag or laminar drag. Use dimensional analysis to determine the formula that describes this drag force (there will be a dimensionless constant left undetermined). [ µ ]=M/(LT), as the SI units of dynamic viscosity are Pa · s or kg/(m · s). Suppose we know that an object with A =3.0
× 10
− 4 m
2 moving through molasses (very viscous µ ≈ 7 Pa · s) at a speed of v =0.50 m/s experiences a drag force of F d
=2.0
× 10
− 2 object with A =6.0
× 10
− 4 m
2 moving through molasses at a speed of
N. What is the drag force on an v =1.0 m/s?
(b) If an object is moving at high speed through a material with low viscosity the only parameters that can affect F d are the object’s speed ( v ), its cross-sectional area ( A ), and the fluid’s density
( ρ ).
This limit is known as Newtonian drag or turbulent drag.
Use dimensional analysis to determine the formula that describes this drag force (there will be a dimensionless constant left undetermined). [ ρ ]=M/L 3 , as the SI units of density are kg/m 3 . Suppose we know that an object with A =3.0
× 10
− 4
F d
=4.4
× 10
− 4 m 2 moving through air at a speed of v =2.50 m/s experiences a drag force of
N. What is the drag force on an object with A =6.0
× 10
− 4 m 2 moving through air at a speed of v =5.0 m/s?

SP221/2141&4341 Chapters 1 & 2 Homework Due Thursday September 8, 2016
Question 3: At time t =0, an object is located at x =0 m and has an initial velocity of v
0 x
=0 m/s.
Starting at t =0, the object accelerates in one dimension according to the time-dependent acceleration a x
( t ) = a
0 e
− t/τ .
[10 points]
(a) Using the constants a
0 and τ , determine the formulas that describe the velocity ( v x
( t )) and position ( x ( t )) of this object as a function of time. (b) Assume that the initial acceleration is given by a
0
=3.50 m/s 2 and that the time constant is given by velocity, and position of this object at time t =5.00 s.
τ =2.500 s. What is the acceleration,
Question 4: At time t =0, an object is located at x =0 m. Starting at t =0, the object moves in one dimension with a velocity given by v x
( t ) = v
0
τ t + τ
. [10 points]
(a) Using the constants v
0 and τ , determine the formulas that describe the acceleration ( a x
( t )) and position ( x ( t )) of this object as a function of time. (b) Assume that the initial velocity is given by v
0
=3.50 m/s and that the time scale is given by τ =2.00 s. What is the acceleration, velocity, and position of this object at time t =4.00 s.
PROGRAMMING PROBLEM
Question 5: A particle is constrained to move along the x -axis. At time t =0 it is at rest and located at the origin [i.e.
x (0) = 0 and v x
(0) = 0]. Starting at t =0 it is subject to a time-dependent acceleration given by a x
( t ) = t
2
(1+2 t ) 3 / 2 ln( t ) , where a x has units of m/s 2 when t has units of seconds. Plot the particle’s acceleration, velocity, and position as functions of time for 0 ≤ t ≤ 5 s.
[20 points]
I will provide an example Matlab program that solves a similar problem.
You can build off of my code any way you wish. Turn in a print out of your computer program as well as the three requested plots. Do not forget to accurately label all axes in your plots. NOTE: In Matlab, the natural log of x is coded as simply log(x) while the base-10 logarithm of x is coded as log10(x) .