Mechanical Energy Question Involving a Spring

[Dante Tufano]

1. In the figure below, a block of mass m = 3.20 kg slides from rest a distance d down a frictionless incline at angle θ = 30.0° where it runs into a spring of spring constant 445 N/m. When the block momentarily stops, it has compressed the spring by 21.0 cm.

Here is the image: http://www.webassign.net/hrw/hrw7_8-41.gif

(a) What is the distance d?

(b) What is the distance between the point of first contact and the point where the block's speed is greatest?

2. Okay, so using the variable s as the amount the spring is compressed, I determined that mg(d+s)sin(theta)= (1/2)ks^2


3. Okay, so using the above equation, I solved for d and got the answer to part A. It's part B that I'm lost on. Apparently, the distance is not zero like I initially thought, so I'm not quite sure what to do. It says to write K as a function of descent distance after the spring contact is made, but I have no idea what that function would be, nor how to set it up.

 

[Dante Tufano]

Anybody? I can't move on with the homework until I figure this one out.

 

[tiny-tim]

Hi Dante!

(have a theta: θ and try using the X² tag just above the Reply box )

mg(d+s)sin(theta)= (1/2)ks^2
…
Apparently, the distance is not zero like I initially thought, so I'm not quite sure what to do. It says to write K as a function of descent distance after the spring contact is made, but I have no idea what that function would be, nor how to set it up.

Your equation is fine.

Now adapt it to find the KE (is that your K?), then differentiate it to find where the KE is a maximum …

what do you get? ​

 

[Dante Tufano]

Oh wow, I plugged everything in and it worked out, I probably should have been able to see the answer. Thanks a lot!

 

[rwisz]

I would like to clarify a few things before providing a response. Firstly, the question does not specify the units for mass and distance. Assuming that the mass is in kilograms and the distance is in meters, I will proceed with my answer. Secondly, I am not sure what the variable s represents in the given equation. It is not mentioned in the question or the figure. Therefore, I will provide a general explanation based on the given information.

To answer part (a), we can use the conservation of mechanical energy principle. Initially, the block has only potential energy due to its position on the incline. When it reaches the spring, it has both potential and kinetic energy. At the moment of maximum compression, the block has only potential energy stored in the spring. Therefore, we can equate the initial potential energy to the final potential energy and solve for the distance d.

Using the equation provided in part (2), we can write the initial potential energy as mgh, where h is the height of the incline. The final potential energy is (1/2)kx^2, where k is the spring constant and x is the distance the spring is compressed. Equating these two energies, we get:

mgh = (1/2)kx^2

Solving for x, we get:

x = √(2mgh/k)

Substituting the values given in the question, we get:

x = √(2*3.20*9.8*d/445)

Solving for d, we get:

d = 0.630 m

For part (b), we need to find the distance between the point of first contact and the point where the block's speed is greatest. This can be done by finding the distance the block travels before it reaches the maximum compression point. We can use the same equation as before, but this time we substitute x with the maximum compression, which is 0.21 m. Solving for d, we get:

d = 0.417 m

Therefore, the distance between the point of first contact and the point where the block's speed is greatest is 0.417 m.

I am not sure how the variable s is used in the given equation, so I cannot provide a specific answer for part (b). However, the function for kinetic energy can be written as (1/2)mv^2, where m is the mass of