In an amusement park ride a cart of mass 300 kg and carrying four passengers each of mass 60 kg

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In an amusement park ride a cart of mass 300 kg and carrying...

In an amusement park ride a cart of mass $300 \mathrm{kg}$ and carrying four passengers each of mass $60 \mathrm{kg}$ is dropped from a vertical height of $120 \mathrm{m}$ along a frictionless path that leads into a loop-the-loop machine of radius $30 \mathrm{m}$ The cart then enters a straight stretch from $A$ to C where friction brings it to rest after a distance of $40 \mathrm{m} \text { . (See Figure } 8.19 .)$ (a) Find the velocity of the cart at A. (b) Find the reaction force from the seat of the cart onto a passenger at $\mathrm{B}$. (c) What is the acceleration experienced by the cart from A to C (assumed constant)?

In an amusement park ride a cart of mass $300 \mathrm{kg}$ and carrying four passengers each of mass $60 \mathrm{kg}$ is dropped from a vertical height of $120 \mathrm{m}$ along a frictionless path that leads into a loop-the-loop machine of radius $30 \mathrm{m}$ The cart then enters a straight stretch from $A$ to C where friction brings it to rest after a distance of $40 \mathrm{m} \text { . (See Figure } 8.19 .)$
(a) Find the velocity of the cart at A.
(b) Find the reaction force from the seat of the
cart onto a passenger at $\mathrm{B}$.
(c) What is the acceleration experienced by the cart from A to C (assumed constant)?

Key Concepts

Normal Force in Circular Motion

The normal force in circular motion is the reaction force exerted perpendicular to the surface of contact. In situations where an object is moving along a curved path, the normal force may vary to provide the necessary centripetal force. This concept is important for understanding how forces interact with objects in vertical circular motions, influencing safety and design in systems like roller coaster loops.

Uniform Acceleration

Uniform acceleration refers to a constant rate of change of velocity over time or distance. When friction or another constant force is applied over a known distance, kinematic equations can be used to calculate parameters such as acceleration or deceleration. This concept is essential for analyzing motions where an object comes to rest, allowing the determination of stopping distances or times.

Conservation of Mechanical Energy

This principle states that in the absence of non-conservative forces (like friction), the total mechanical energy of a system remains constant. In problems involving vertical drops or inclines, gravitational potential energy is converted into kinetic energy, allowing the determination of velocities at different points along the path.

Centripetal Motion

Centripetal motion refers to the movement of an object along a circular path, where a net inward force is required to keep the object moving in a circle. It is characterized by centripetal acceleration, which is given by v²/r, and is crucial for analyzing forces in scenarios such as loops or curved trajectories in mechanical systems.

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