ME 101:
Measurement Demonstration (MD3)
Vectors, Statics and Dynamics Measurements
Mechanical Engineering 101
University of Kentucky
Reminders / Homework
• Homework #3 – Due Wednesday 10/6 - Today
– Wickert - 3.2, 3.4, 3.5, 3.13
• Computer Demonstration 2 (CD2) – Due Friday 10/8
• Career Fair – Wednesday 10/13
– http://www.uky.edu/CareerCenter/
– Student Center Grand Ballroom
– 12:00 PM – 4:00 PM
– 5 Extra Credit Points
• 3 Business Cards (Stapled to a sheet of paper)
Mechanical Engineering 101
University of Kentucky
Introduction
Purpose:
To examine spring-mass systems
– Determine spring forces
– Determine deformations
– Determine spring constants
– Drawing FBD’s
– Simple harmonic motion
Mechanical Engineering 101
University of Kentucky
Experiment
Experiment Materials
• Two springs
– One large
– One small
– Different spring constants
• Weight Basket
• Weights/Masses
Experiment Tools
• Linear Scale (Use cm Scale)
• Scale (Grams)
• Protractor (Degrees)
• Stopwatch (Seconds)
Mechanical Engineering 101
University of Kentucky
Problem
• A mass hangs from a spring
• Causes stretch in the spring
• Displaced a distance d from its
unstretched length
Unstretched
Length
• How do we describe this system?
Equilibrium
Stretch (d)
W
Equilibrium
Position
Mechanical Engineering 101
University of Kentucky
Hooke’s Law
Springs are governed by Hooke’s law:
F  kx
where:
• F is the force (or weight) applied to the spring
• x is the distance displaced
• k is the spring constant
Mechanical Engineering 101
University of Kentucky
Static Behavior
• A mass of known weight hangs
from a spring
F  kx
– Allow system to come to rest
• What is the spring constant k?
• How do you solve problem?
W
x
Mechanical Engineering 101
University of Kentucky
Static Behavior
• Step 1: Draw a Free Body Diagram
• Step 2: Set the force equal to the
weight of the mass
• Step 3: Measure the displacement
• Step 4: Solve for k
F  kx
W
x
Mechanical Engineering 101
University of Kentucky
Static Behavior: 2 Springs
What is the deflection with two springs?
k1
k2
1 1
F     x
 k1 k2 
Mechanical Engineering 101
University of Kentucky
Static Behavior: 2 Springs
What are the forces in the springs?
k1
k2
W
F = (k1 + k2) ∆x
Mechanical Engineering 101
University of Kentucky
Static Behavior: 2 Springs
• Step 1: Draw a Free Body Diagram
• Step 2: Set the force equal to the weight of the mass
• Step 3: Measure the displacement
• Step 4: Solve for k
F1
F = (k1 + k2) ∆x
F2
W
Mechanical Engineering 101
University of Kentucky
Dynamic Behavior
• A block hangs from a spring
• Pull down 10 cm and release
• Bounces up and down at a rate of X
times every second
• How could this rate be increased?
Mechanical Engineering 101
University of Kentucky
Dynamic Behavior
You may choose any of:
– increase the mass
– decrease the mass
Which is correct?
– push it faster
– make the spring stiffer
– make the spring less stiff
– pull it down farther
– don’t pull it down as far
– other?
Mechanical Engineering 101
University of Kentucky
Spring Constant & Period
• The period T (or frequency f) of an oscillator is
related only to:
– the stiffness (k) of the restoring force
– the inertia (m) of the system
• It does not tell you anything about how the system
got started
• The frequency is independent of the amplitude
Mechanical Engineering 101
University of Kentucky
Spring Constant & Period
T  2 m
1
f  1/ T 
2
k
k
m
n  2f  k m
Mechanical Engineering 101
University of Kentucky
Problem
A 2.0 kg block hangs from a spring and oscillates with a
period T = 1 s
What mass of block, hung from the same spring, would
have T = 2 s?
T  2
m
k
Therefore, m = 8.0 kg
Mechanical Engineering 101
University of Kentucky
Key Concepts
• The spring constant can be found from Hooke’s law
or the period
• Static equilibrium means the sum of the forces = 0
• Dynamic motion results when sum of the forces ≠ 0
• The period T depends on k and m only
Mechanical Engineering 101
University of Kentucky
Resonance
• Tacoma Narrows Bridge
• We’ll see this in later labs!
Mechanical Engineering 101
University of Kentucky
Lab Procedure – Part I – Spring Calibration
1. Select one of the springs provided to you. Make sure you indicate
which spring it is so you can keep track of it during the experiment.
2. Hang the spring vertically and measure the location of the tip. This is
your starting point for that spring.
3. Weigh the basket and hang from the spring. Add weights to the basket
until it deflects. Wait for the system to come to rest. Measure the
deflection from the zero point and record it as your offset. Record the
deflection in cm (x) and weight (F) in Newtons. This is the first point on
your curve.
4. Increment the weight attached to the spring by adding a small mass.
Make sure the system comes to rest. Measure and record the
deflection and total applied force. Repeat 4 times.
5. Plot all 5 points in Excel. Using the curve fitting function, calculate
the slope of the curve to determine the spring constant k.
6. Repeat for the other spring. Make sure to include your table and plots
for each spring (1 page for each spring).
Mechanical Engineering 101
University of Kentucky
Lab Procedure – Part II – Static Measurements
1. Attach one end of each of the springs to the table, 30 cm (12 inches)
apart.
2. Attach the mass hanger to the end of the two springs. Add weights to the
basket until both springs begin to deflect. Wait for the system to come to
rest. This will be your zero point. Draw the FBD of the system.
3. Measure the angle of the springs with respect to the horizontal plane and
the displacement of the springs.
4. Find the magnitude F and direction for each of the vectors. Write
them in equation form (F = Fxi+Fyj).
5. Determine the net force in each direction (Fx and Fy). Are they
equal to zero? Should they be?
6. Repeat steps 3-5 for 4 other loads on the system.
Mechanical Engineering 101
University of Kentucky
Lab Procedure – Part III – Dynamic Measurements
1. Using the larger of the two springs, hang it vertically and attach a mass
of approximately 140 grams (14 N) to the spring.
2. Pull the spring down a measurable distance and allow it to oscillate.
3. Using the stopwatch, record the number of times the system oscillates
in 30 seconds.
4. Calculate the period and frequency of the spring-mass system.
5. Using the equation for frequency and the known parameters of the
system, calculate the spring constant. Compare this value with
that measured in Part I.
Mechanical Engineering 101
University of Kentucky
Reminders / Homework
• Homework #4 – Due Wednesday 10/13
– Wickert - 3.15, 3.17, 3.29, 3.30
• Computer Demonstration (CD2) – Due Friday 10/8
• Career Fair – Wednesday 10/13
– http://www.uky.edu/CareerCenter/
– Student Center Grand Ballroom
– 12:00 PM – 4:00 PM
– 5 Extra Credit Points
• 3 Business Cards (Stapled to a sheet of paper)
• Measurement Demonstration (MD3) – Friday 10/8
– Attendance is mandatory, no make-ups will be given
– Get to class early if possible, we will start on time it takes the full 50 minutes
– Same team assignments as MD2 (Posted on BlackBoard)
Mechanical Engineering 101
University of Kentucky