In Fig. 8-42, a block of mass m=3.20 kg slides from rest a distance d down a frictionless incli

step 1 In this problem on the topic of conservation of energy, we have a block with mass 3 .2 kilograms

In Fig. 8-42, a block of mass m=3.20 kg slides from rest a...

In Fig. 8-42, a block of mass $m=3.20 \mathrm{~kg}$ slides from rest a distance $d$ down a frictionless incline at angle $\theta=30.0^{\circ}$ where it runs into a spring of spring constant $431 \mathrm{~N} / \mathrm{m}$. When the block momentarily stops, it has compressed the spring by $21.0 \mathrm{~cm} .$ What are (a) distance $d$ and (b) the distance between the point of the first block-spring contact and the point where the block's speed is greatest?

In Fig. 8-42, a block of mass $m=3.20 \mathrm{~kg}$ slides from rest a distance $d$ down a frictionless incline at angle $\theta=30.0^{\circ}$ where it runs into a spring of spring constant $431 \mathrm{~N} / \mathrm{m}$. When the block momentarily stops, it has compressed the spring by $21.0 \mathrm{~cm} .$ What are
(a) distance $d$ and (b) the distance between the point of the first block-spring contact and the point where the block's speed is greatest?

Key Concepts

Conservation of Mechanical Energy

In a frictionless system, the total mechanical energy is conserved. This means that the loss in gravitational potential energy as an object moves along an incline is exactly equal to the gain in other forms of energy, such as the elastic potential energy stored in a spring or the kinetic energy of the object. This principle allows one to set up equations equating the energy lost to the energy gained, which is essential in solving problems that involve motion and spring compression.

Gravitational Potential Energy on an Incline

When an object is displaced along an incline, the vertical change in height is less than the distance traveled along the incline and is related by the sine of the angle of the incline. The gravitational potential energy change is given by the product of mass, gravitational acceleration, and the vertical displacement, which is calculated as the product of the distance along the incline and the sine of the angle. This concept is key to linking the motion along the incline to changes in energy.

Elastic Potential Energy

A spring stores energy when it is compressed or stretched, and the elastic potential energy is given by half the product of the spring constant and the square of the displacement from its equilibrium position. This energy storage mechanism, analogous to gravitational potential energy, is crucial for analyzing systems where objects interact with springs, as it provides the basis for determining how far a spring is compressed or extended as energy is transferred.

Energy Conversion Equations

By applying the conservation of mechanical energy, one can equate the gravitational potential energy lost by an object moving down an incline to the elastic potential energy gained by a spring when it is compressed. This method allows the derivation of relationships and equations that link physical quantities such as mass, gravitational acceleration, incline angle, distance traveled, spring constant, and compression distance, enabling problem solving in mechanics.

Force Equilibrium and Maximum Speed

In systems where a mass engages with a spring, the point of maximum speed is not necessarily at the beginning or end of the spring compression, but occurs where the net force acting on the mass becomes zero. This happens when the force due to gravity along the incline is exactly balanced by the spring force. Understanding and analyzing the equilibrium of forces, using Newton’s second law, is essential to determining the location along the path where the object reaches its maximum velocity.

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