Conservation Of Energy, Block Sliding Down Incline Onto Spring.

In summary, the conversation discusses a problem involving a block of mass m sliding down a frictionless plane and striking a spring of force constant k. The goal is to find the distance the spring is compressed when the block momentarily stops. The solution involves using conservation of energy equations and solving a quadratic formula to find the value of x.
  • #1

Homework Statement


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A block of mass m starts from rest at a height h and slides down a frictionless plane inclined at angle θ with the horizontal, as shown below. The block strikes a spring of force constant k.
Find the distance the spring is compressed when the block momentarily stops. (Let the distance the block slides before striking the spring be l. Use the following as necessary: m, θ, k, l, and g.)

Homework Equations


Conservation of Energy:
GPE : U = mgh
K = (1/2)mv2
Spring = (1/2)kx2


The Attempt at a Solution



My Diagram:
QJ9jkix.jpg

Let H = l*sinθ
Let h = x*sinθ

Moment 1: E = mg(H+h) = mgsinθ(l+ x)

Moment 2: E = (1/2)mv2 + mgh = (1/2)mv2 + mgx*sinθ

Moment 3: E = (1/2)kx2

-------- Attempts to get rid of x2 led to
x = h/sin
(1/2)k(h/sinθ)2 = mgsinθ(l+ x) ;;;
(kh2/2sin2θ) = mgsinθ(l+ x) ;;;
(kh2/2) = mglsin3θ + mgxsin3θ ;;;
(kh2/2) - mglsin3θ = mgxsin3θ ;;;

x = k*h2/2mgsin3θ - l
 
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  • #2
Creepypunguy said:
Moment 1: E = mg(H+h) = mgsinθ(l+ x)

Moment 2: E = (1/2)mv2 + mgh = (1/2)mv2 + mgx*sinθ

Moment 3: E = (1/2)kx2
This is good.

-------- Attempts to get rid of x2 led to
x = h/sin
(1/2)k(h/sinθ)2 = mgsinθ(l+ x) ;;;
(kh2/2sin2θ) = mgsinθ(l+ x) ;;;
(kh2/2) = mglsin3θ + mgxsin3θ ;;;
(kh2/2) - mglsin3θ = mgxsin3θ ;;;

x = k*h2/2mgsin3θ - l
Don't try to express things in terms of "h"--that's an unknown directly related to "x".

Hint: Just compare moments 1 and 3. Solve that equation!
 
  • #3
As h=xsinθ, your last equation does not give x explicitly in terms of the known quantities, k, m, θ, L, and g.

ehild
 
  • #4
So by solving equations 1 and 3 I got
mglsinθ+mgxsinθ = (1/2)kx2
0 = (k/2)x2 - mgsinθx - mglsinθ
quadratic formula

x = (mgsinθ ± √[mgsinθ(mgsinθ+2kl)] ) / k
 
  • #5
It is all right now.

ehild
 

What is conservation of energy?

Conservation of energy is a fundamental principle in physics that states energy cannot be created or destroyed, but can only be converted from one form to another.

How does conservation of energy apply to a block sliding down an incline onto a spring?

In this scenario, the potential energy of the block at the top of the incline is converted into kinetic energy as it slides down. When the block reaches the bottom and compresses the spring, the kinetic energy is converted into elastic potential energy. The total energy of the system (block and spring) remains constant throughout the process.

What factors affect the conservation of energy in this scenario?

The conservation of energy in this scenario is affected by the mass of the block, the angle of the incline, and the stiffness of the spring. These factors determine the amount of potential and kinetic energy present in the system.

What is the equation for calculating the potential energy of the block?

The potential energy of the block can be calculated using the equation PE = mgh, where m is the mass of the block, g is the acceleration due to gravity, and h is the height of the block on the incline.

How can conservation of energy be applied to real-world situations?

Conservation of energy is a fundamental principle that applies to all physical systems. It can be used to analyze and understand real-world situations such as the motion of objects, electricity and magnetism, and chemical reactions. It also plays a crucial role in the development of sustainable energy sources and efficient energy use.

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