The roller-coaster track shown is contained in a vertical...

The roller-coaster track shown is contained in a vertical plane. The portion of track between $A$ and $B$ is straight and horizontal, while the portions to the left of $A$ and to the right of $B$ have radii of curvature as indicated. A car is traveling at a speed of $72 \mathrm{km} / \mathrm{h}$ when the brakes are suddenly applied, causing the wheels of the car to slide on the track $\left(\mu_{k}=0.20\right) .$ Determine the initial deceleration of the car if the brakes are applied as the car $(a)$ has almost reached $A,$ (b) is traveling between $A$ and $B,(c)$ has just passed $B .$

The roller-coaster track shown is contained in a vertical plane. The
portion of track between $A$ and $B$ is straight and horizontal, while
the portions to the left of $A$ and to the right of $B$ have radii of curvature as indicated. A car is traveling at a speed of $72 \mathrm{km} / \mathrm{h}$ when
the brakes are suddenly applied, causing the wheels of the car to
slide on the track $\left(\mu_{k}=0.20\right) .$ Determine the initial deceleration of the car if the brakes are applied as the car $(a)$ has almost reached $A,$
(b) is traveling between $A$ and $B,(c)$ has just passed $B .$

Key Concepts

Centripetal Acceleration

Centripetal acceleration is the acceleration that keeps an object moving in a circular path, directed towards the center of the curvature. In problems involving curved motion, it is important to recognize that any net force causing this acceleration must be accounted for along with tangential forces, especially when friction is the sole source of other acceleration along the path.

Force Resolution in Curved Motion

When an object moves along a curved path, forces must be decomposed into components tangent and normal (radial) to the path. This resolution is crucial when the friction force, which acts along the direction of the relative motion, is not aligned with the tangent to the path or must be split between providing tangential deceleration and centripetal acceleration.

Kinetic Friction

Kinetic friction is the resistive force that occurs when two surfaces slide against each other. In dynamics problems, this frictional force is calculated as the product of the coefficient of kinetic friction and the normal force, and it acts in a direction opposite to the motion, contributing to the object’s deceleration.

Newton's Second Law

Newton's Second Law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma), is fundamental in analyzing the motion of objects. It allows us to relate the frictional force and other forces present to the resulting deceleration of the moving object.

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