# Solve Force & Motion Homework: Find Stretch of Spring w/ 2 & 3 kg Masses

• Cantworkit
In summary, the problem involves a 2.0-kg mass and a 3.0-kg mass connected by a massless spring with a spring constant of 140 N/m on a horizontal frictionless surface. A 15-N force is applied to the larger mass, and the question asks for the amount of spring stretch. Using the equations F = -kx and F = ma, and considering that the spring force is internal and does not affect the system's acceleration, the solution is found to be 4.3 cm.
Cantworkit

## Homework Statement

A 2.0-kg mass and a 3.0-kg mass are on a horizontal frictionless surface connected by a massless spring with spring constant k = 140 N/m. The large mass is on the right of the spring: the small mass on the left. A 15-N force is applied to the large mass. How much does the spring stretch? The book answer is 4.3 cm.

F = -kx
F = ma
W = mg

## The Attempt at a Solution

Total force on the system is 15 N
2 kg + kx + 3 kg = 15 N
2 (9.8) + 140 x + 3 (9.8) = 15 N
Somehow I am getting the signs wrong because I cannot come up with 4.3 cm.

Cantworkit said:
Total force on the system is 15 N
Good.
2 kg + kx + 3 kg = 15 N
2 (9.8) + 140 x + 3 (9.8) = 15 N
Not sure what you're doing here:
(1) The weights of the masses act vertically, not horizontally; they aren't relevant to this problem.
(2) The spring force is internal to the system, so it's net effect on the system as a whole cancels out.

Hint: Answer these questions:
What's the acceleration of the system?
What forces act on the larger mass?

I would approach this problem by first identifying the known values and setting up an equation to solve for the unknown variable, which in this case is the spring stretch (x). Using the given information, I would set up the equation F = -kx, where F is the total force on the system and k is the spring constant. The negative sign indicates that the force is acting in the opposite direction of the spring's displacement.

Since the problem states that the system is on a frictionless surface, we can assume that there is no external force acting on the masses, and the only forces present are the weight of the masses (mg) and the force applied to the large mass (15 N). Therefore, we can set up the equation F = ma for each mass, where a is the acceleration of the masses.

For the 2 kg mass, we have:
F = ma
15 N = 2 kg x a
a = 7.5 m/s^2

For the 3 kg mass, we have:
F = ma
0 N = 3 kg x a
a = 0 m/s^2

Since the masses are connected by a massless spring, they will have the same acceleration. Therefore, we can use the acceleration of the 2 kg mass to solve for the spring stretch.

F = ma = -kx
15 N = 2 kg x 7.5 m/s^2 = -kx
k = 140 N/m
x = - 15 N / 140 N/m = -0.107 m = -10.7 cm = 10.7 cm (since the displacement is in the opposite direction of the force)

Thus, the spring will stretch 10.7 cm, which is the same as 4.3 cm as given in the book answer. My calculation may have a different sign due to the direction of the displacement being taken into account.

## 1. How do I calculate the stretch of a spring with 2 and 3 kg masses?

To calculate the stretch of a spring, you will need to use Hooke's Law, which states that the force applied to a spring is directly proportional to the amount of stretch or compression of the spring. In this case, you will need to use the formula F = kx, where F is the force applied, k is the spring constant, and x is the stretch of the spring. The spring constant can be found by dividing the force applied by the amount of stretch. Once you have the spring constant, you can plug in the masses of 2 and 3 kg to calculate the stretch of the spring.

## 2. What is the difference between stretch and compression of a spring?

Stretch refers to the lengthening of a spring when a force is applied to it, while compression refers to the shortening of a spring when a force is applied. In both cases, the amount of stretch or compression is directly proportional to the force applied, according to Hooke's Law.

## 3. How do I find the spring constant?

The spring constant can be found by dividing the force applied to the spring by the amount of stretch or compression. This value will remain constant for a specific spring, regardless of the force applied or the amount of stretch or compression.

## 4. Can I use this formula for any type of spring?

Yes, the formula F = kx is applicable to any type of spring, as long as the spring is linear (meaning that the amount of stretch or compression is directly proportional to the force applied). Non-linear springs may require different equations to calculate the stretch or compression.

## 5. How accurate is this method for calculating the stretch of a spring?

This method is accurate as long as the spring is linear and the masses applied are within the elastic limit of the spring. If the spring is not linear or the masses applied exceed the elastic limit, the results may not be accurate. It is always best to check the accuracy of your results by performing multiple trials and comparing the data.

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