Conservación de la energía | problema esfera y resorte
Energy Conservation Problem: Spring Compression
Introduction to the Problem
- The problem involves an 8 kg mass released from rest on a smooth curved ramp, compressing a spring with a constant K = 400 , textN/m at the bottom. The goal is to calculate the deformation of the spring when the mass compresses it.
Key Points in Energy Conservation
- Two critical points are identified: Point 1 (top of the ramp) and Point 2 (where the mass compresses the spring). At Point 1, velocity is zero and height is 10 m; at Point 2, both velocity and height are zero.
- The initial conditions include:
- Velocity at Point 1: v_1 = 0
- Height at Point 1: h_1 = 10 , m
- Velocity at Point 2: v_2 = 0
- Height at Point 2: h_2 = 0
Energy Types Involved
- Kinetic Energy ( KE ): Defined as KE = 1/2 mv^2 , where m is mass and v is velocity.
- Gravitational Potential Energy ( PE_g ): Given by PE_g = mgh, where g =9.81,m/s^2.
- Elastic Potential Energy ( PE_e ): For springs, defined as PE_e = 1/2 kx^2, where k is spring constant and x is compression distance.
Application of Conservation of Energy Principle
- The principle states that energy in a closed system remains constant if no dissipative forces act on it. Thus:
[ E_point,1 = E_point,2 + E_elastic.]
This translates to:
[ PE_g + KE_g |_point,1 = PE_g + KE_g |_point,2 + PE_e |_spring.[]]
Solving for Spring Deformation
- Setting up equations based on conservation principles leads to:
[ mgh_1 + KE_point,1 = KE_point,2 + PE_e.]
Given that velocities are zero at both points simplifies this to:
[ mgh_1 = PE_e.]
Substituting known values yields:
[ (8 kg)(10 m/s^2)(10 m) = (400 N/m)left(x^2/2right).]
Solving gives us:
[ x^3 =4,]
thus,
[ x=sqrt4=2m.]
The deformation of the spring is therefore determined to be ** x=2m**.