Analyze a spring-mass system consisting of a combination of two or more springs attached to an object with either a side-by-side configuration or an end-to-end configuration.
A 56.5-gram mass is hanging on the end of a spring with spring constant 2.8 N/m as shown in situation B at right. Consider an identical spring helping to hold up the mass as shown in situation C.
What is the stretch, X1?
Stretch X1
cm
How much weight does each spring support in situation C?
Weight per Spring
Newtons
What is the stretch, X2?
Stretch X2
Suppose you replace the 2 springs in situation C holding up the mass with one spring that results in the same stretch, X2. What would its spring constant need to be?
1 Equivalent Spring Const
N/m
By what factor would the angular frequency change going from one spring in situation B to 2 identical springs side-by-side in situation C?
Angular Frequency Factor
By what factor would the cyclic frequency change going from one spring in situation B to 2 identical springs side-by-side in situation C?
Cyclic Frequency Factor
By what factor would the period change going from one spring in situation B to 2 identical springs side-by-side in situation C?
Period Factor
A 91.5-gram mass is hanging on the end of a spring with spring constant 4.0 N/m as shown in situation B at right. Consider an identical spring connected to the first to hold up the mass as shown in situation D.
How much weight does each spring support in situation D?
Suppose you replace the 2 springs in situation D holding up the mass with one spring that results in the same stretch, X2. What would its spring constant need to be?
Spring Constant
By what factor would the angular frequency change going from one spring in situation B to 2 identical springs in series in situation D?
By what factor would the cyclic frequency change going from one spring in situation B to 2 identical springs in series in situation D?
Cyclic Frequency
By what factor would the period change going from one spring in situation B to 2 identical springs in series in situation D?
A 87.0-gram mass is hanging on the end of a spring with stretch X1 measured to be 6.0 cm as shown in situation B at right. Consider a second spring added such that X2 is measured to be 25.0 cm as shown in situation D.
What is the spring constant for the spring in situation B?
What is the spring constant for the 2-spring setup in situation D?
Equiv Spring Constant
What is the spring constant for the added spring in situation D?
What is the ratio of the angular frequency for the 2-spring system in situation D over the 1-spring system in situation B?
Ang Freq Ratio: D over B
A 86.5-gram mass is attached to the end of a spring with spring constant 1.66 N/m as shown in situation B at right. Consider an identical spring attached to the mass as shown in situation A. Consider both situations in A and B pulled to the right a distance A and released.
What is the angular frequency of the spring-mass system in situation B?
Angular Frequency
rad/sec
What is the angular frequency of the spring-mass system in situation A?
By what factor does the maximum Potential Energy change when the 2nd spring is added?
Max Spring PE Factor
By what factor does the maximum Kinetic Energy change when the 2nd spring is added?
Max Kinetic Energy Factor
By what factor does the maximum speed change when the 2nd spring is added?
Maximum Speed Factor
By what factor does the magnitude of the maximum acceleration change when the 2nd spring is added?
Max Acceleration Factor
A 118-gram mass is attached to the end of a spring with spring constant 18.3 N/m and pulled a distance of X1 as shown in situation C at right. The force to hold the mass against the spring at this position is 3.6 Newtons. Consider adding a spring with spring constant 29.1 N/m in series with the first spring and pulling the mass a distance of X2 as shown in situation A at right. The force to do this is also 3.6 Newtons. In both situations A and C the mass is released and oscillates back and forth.
What is the stretch X1 of the spring-mass system in situation C?
What is the stored energy of the spring-mass system in situation C?
Spring PE
Joules
What is the same for both springs when stretched in situation A?
What is the stretch of the original spring in the spring-mass system in situation A?
Original Spring Stretch
What is the stretch of the added spring in the spring-mass system in situation A?
Added Spring Stretch
What is the stretch X2 of the spring-mass system in situation A?
What would be the spring constant of a spring that could replace the 2 spring combination in situation A and provide the same behavior?
Equivalent Spring Const
What is the stored energy of the original spring in the spring-mass system in situation A?
What is the stored energy of the added spring in the spring-mass system in situation A?
What is the total stored energy in the spring-mass system in situation A?
Total Spring PE
What is the ratio of the maximum Kinetic Energy in situation A over the maximum Kinetic Energy in situation C?
Max KE Ratio: A to C
What is the ratio of the maximum speed in situation A over the maximum speed in situation C?
Max Speed Ratio: A to C
What is the ratio of the angular frequency in situation A over the angular frequency in situation C?
Ang Freq Factor: A to C
A 500-gram mass is attached to two relaxed springs in the horizontal direction as shown at right in situation B. The left spring has a spring constant of 6.2 N/m and the right spring has a spring constant of 13.9 N/m. The mass is displaced to the right a distance of A as shown in situation A. The force exerted by the left spring at this position is 0.74 Newtons.
When an outside force displaces the mass to the right of the relaxed position, what direction is the left spring's force on the mass?
When an outside force displaces the mass to the right of the relaxed position, what direction is the right spring's force on the mass?
What is the same for both springs in situation A?
What is the displacement of the mass, A, in situation A?
Displacement, A
What is the force put on the mass by the spring on the right?
Right Spring Force
What is the net force on the mass from the 2 springs?
Spring Net Force
What is the angular frequency of this spring-mass system?
What is the maximum speed if the mass is released from the position in situation A?
Maximum Speed
cm/sec
What is the maximum acceleration if the mass is released from the position in situation A?
Maximum Acceleration
cm/s/s
What percent of the stored energy in the springs is in the left spring in Situation A?
Percent PE in Left Spring
%