A spring with k=40.0 N / m is at the base of a frictionless 30.0^∘ inclined plane. A 0.50 kg obj

step 1 Hi friends as soon is the figure there is a block of mass 0 .5 kg in contact to the spring of sp

A spring with k=40.0 N / m is at the base of a frictionless...

A spring with $k=40.0 \mathrm{N} / \mathrm{m}$ is at the base of a frictionless $30.0^{\circ}$ inclined plane. A $0.50 \mathrm{kg}$ object is pressed against the spring, compressing it $0.20 \mathrm{m}$ from its equilibrium position. The object is then released. How far along the incline does the object travel before coming to rest and then sliding back down?

A spring with $k=40.0 \mathrm{N} / \mathrm{m}$ is at the base of a frictionless $30.0^{\circ}$ inclined plane. A $0.50 \mathrm{kg}$ object is pressed against the spring, compressing it $0.20 \mathrm{m}$ from its equilibrium position. The object is then released.
How far along the incline does the object travel before coming to rest and then sliding back down?

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Key Concepts

Conservation of Energy

This concept refers to the principle that in a closed system with only conservative forces (such as gravitational and elastic forces), the total mechanical energy remains constant. In problems like this one, energy stored in one form (spring potential energy) is transformed into another form (gravitational potential energy) without any losses, allowing for the calculation of distances or speeds based on energy balances.

Spring Potential Energy and Hooke’s Law

Spring potential energy is the energy stored when a spring is compressed or stretched, given by the formula U = 1/2 k x², where k is the spring constant and x is the displacement from equilibrium. This energy is released when the spring returns to its equilibrium position, and is a key concept in analyzing systems where elastic forces are at work.

Gravitational Potential Energy on an Incline

Gravitational potential energy is the energy an object gains due to its position in a gravitational field, typically expressed as U = mgh. On an inclined plane, the vertical height gained is related to the distance traveled along the incline and the sine of the incline angle. This concept is crucial for understanding how the energy from the spring is converted into gravitational potential energy as the object moves upward along the plane.

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