(II) A pinball machine uses a spring launcher that is compressed 6.0 cm to launch a ball up a 15

step 1 So we want to find the spring constant of the spring launcher. Again, we know that we're going t

(II) A pinball machine uses a spring launcher that is...

(II) A pinball machine uses a spring launcher that is compressed 6.0 $\mathrm{cm}$ to launch a ball up a $15^{\circ}$ ramp. Assume that the pinball is a solid uniform sphere of radius $r=1.0 \mathrm{cm}$ and mass $m=25 \mathrm{g}$ . If it is rolling without slipping at a speed of 3.0 $\mathrm{m} / \mathrm{s}$ when it leaves the launcher, what is the spring constant of the spring launcher?

Key Concepts

Conservation of Energy

This principle states that energy cannot be created or destroyed, only transformed from one form to another. In the context of a spring launcher, the potential energy stored in the compressed spring is completely converted into the kinetic energy of the ball (both translational and rotational) when it leaves the launcher.

Rolling Without Slipping

Rolling without slipping is a condition where the point of contact between a rolling object and the surface is momentarily at rest relative to the surface. This establishes a direct relationship between the translational velocity and the angular velocity (v = r?), which is crucial for connecting the kinetic energy components.

Translational and Rotational Kinetic Energy

A rolling object has both translational kinetic energy (associated with its linear motion) and rotational kinetic energy (associated with its rotation about the center of mass). These forms of energy are calculated as (1/2)mv² and (1/2)I?² respectively, and are both essential when analyzing the energy conversion from the spring.

Spring Potential Energy

The energy stored in a compressed spring is expressed as (1/2)kx², where k is the spring constant and x is the compression distance. This concept is key to determining how much energy is available to be converted into the kinetic energy of the ball.

Moment of Inertia for a Sphere

For a solid uniform sphere, the moment of inertia is given by I = (2/5)mr². This parameter is important for computing the rotational kinetic energy of the sphere, which in turn factors into the total kinetic energy obtained from the spring launch.

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