An object is attached to a hanging unstretched, ideal and massless spring and slowly lowered to its equilibrium position, a distance of 5.3 cm below the starting point. If instead of having been lowered slowly the object was dropped from rest, how far then would it then stretch the spring at maximum elongation (measured from the point it was dropped)?

I got an answer of 12.8cm but this is incorrect, (textbook says 11cm rounded)

lower ... m g = k x ... weight = spring force

drop ... m g y = 1/2 k y^2 ... potential energy at top = work to stretch spring

2 m g = k y ... substituting ... 2 k x = k y ... 2 x = y

Hey, Thanks so much for answering!!

To solve this problem, let's first consider the situation when the object is lowered slowly. In this case, the displacement of the object from the equilibrium position is 5.3 cm.

Now, let's analyze what happens when the object is dropped from rest. When the object is dropped, it will initially have gravitational potential energy which will be converted into elastic potential energy as the object stretches the spring.

At maximum elongation, all the potential energy of the object will be converted into elastic potential energy. Therefore, we can equate the gravitational potential energy with the elastic potential energy.

Gravitational potential energy:
PE_grav = mgh

Elastic potential energy:
PE_elastic = (1/2)kx^2

Where:
m = mass of the object
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height from which the object is dropped (cm)
k = spring constant
x = maximum elongation of the spring (cm)

In this case, the height from which the object is dropped (h) is equal to the initial displacement of the object when it was slowly lowered, which is 5.3 cm.

So, we have:
PE_grav = PE_elastic
mgh = (1/2)kx^2

Simplifying the equation, we get:
mg(5.3) = (1/2)kx^2

Now, let's solve for x.

Given that the spring is ideal and massless, we know that the spring constant (k) is a constant value for the given spring.

It seems there might be an error in your calculation or equation setup. I'll perform the calculations using the rounded value of 11 cm, as mentioned in the textbook.

Using the given values:
m * 9.8 * 5.3 = (1/2)k * 11^2

Let's assume the mass of the object (m) is 1 kg for simplicity.

Simplifying the equation further:
9.8 * 5.3 = (1/2)k * 11^2

50.94 = 605.5k

Now, let's solve for k:
k = 50.94 / 605.5 ≈ 0.0841 N/cm

Using this value of k, we can now find x:

x = sqrt((2 * m * g * h) / k)
= sqrt((2 * 1 * 9.8 * 5.3) / 0.0841)
≈ sqrt(607.8973)
≈ 24.67 cm

Therefore, when the object is dropped from rest, it would stretch the spring at maximum elongation by approximately 24.67 cm, rather than the 12.8 cm you calculated.

Please double-check your calculations and verify the information provided in the textbook to ensure you have the correct values.

To find the maximum elongation of the spring when the object is dropped from rest, we need to consider the conservation of mechanical energy.

When the object is dropped, it initially has gravitational potential energy, which is converted into the potential energy stored in the spring when it stretches.

The gravitational potential energy (PE_grav) can be calculated using the equation: PE_grav = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height from which the object is dropped.

Since the object is dropped from rest, it initially has no kinetic energy, so all of its initial potential energy is converted into potential energy stored in the spring.

When the object reaches its maximum elongation, all of its potential energy is converted into potential energy stored in the spring. At this point, the gravitational potential energy is zero, and the potential energy stored in the spring is given by the equation: PE_spring = (1/2)kx^2, where k is the spring constant and x is the maximum elongation of the spring.

To find x, we can equate the two potential energies: mgh = (1/2)kx^2.

Given:
h = 5.3 cm = 0.053 m (height from which the object is dropped)
k (spring constant) is not provided

To find x, we need to know the spring constant (k). Without that information, we cannot calculate the maximum elongation of the spring accurately.

Please check if the spring constant is provided in the problem statement or refer to the given information to find the spring constant.