Find Initial Separation d for Mass-Spring System: Energy & Force | 4245 N/m

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To find the initial separation d in the mass-spring system, the potential energy of the block after compressing the spring was calculated as 123.584 J using the formula 1/2 kx^2. The vertical distance h was determined using mgh, resulting in a height of 1.011 m. However, the required distance d is along the incline, not vertically. The correct approach involves calculating the hypotenuse using the sine of the incline angle, but the final value of 1.7367 m was still incorrect. The discussion emphasizes the importance of correctly converting vertical distance to the incline distance.
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A 21.4 kg mass starts from rest and slides a distance d down a frictionless 35.6 degree incline where it contacts an uncompressed spring of negligible mass, as shown. The mass slides an additional distance 0.2413 m as it is brought momentarily to rest by compressing the spring (force constant k = 4245 N/m). Find the initial separation d between the mass and the spring.

i used 1/2 kx^2 to find the PE of the block after being compressed to be 123.584 J. then i used mgh to find h. so 123.584/(35.6*21.4) = 1.011 m... but its not right. what did i do wrong? thanks.
 
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You found the vertical distance h that is traversed, what they are asking for the the distance d along the incline it travels. You have the y-component of the distance, now you have to calculate the hypoteneuse which is d.

You did everything right, you just didn't take the final step.
 
so i divided 1.011 by sin 35.6 but it is still not right. (1.7367)
 

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