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Ch 08: Dynamics II: Motion in a Plane
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 08: Dynamics II: Motion in a PlaneProblema 18
Capitolo 8, Problema 18

A car drives over the top of a hill that has a radius of 50 m. What maximum speed can the car have at the top without flying off the road?

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Identify the forces acting on the car at the top of the hill. The two forces are the gravitational force (mg, acting downward) and the normal force (N, also acting downward at the top of the hill). The centripetal force required to keep the car moving in a circular path is provided by the gravitational force and the normal force.
Set up the condition for the car to just stay on the road without flying off. At the maximum speed, the normal force (N) becomes zero, meaning the gravitational force alone provides the centripetal force. Use the equation for centripetal force: Fc = mv2/r, where v is the speed, r is the radius of the hill, and m is the mass of the car.
Equate the gravitational force to the centripetal force at the top of the hill. This gives the equation: mg = mv2/r, where g is the acceleration due to gravity (approximately 9.8 m/s²).
Simplify the equation by canceling out the mass m on both sides, resulting in: g = v2/r. Rearrange to solve for v: v = gr.
Substitute the values for g (9.8 m/s²) and r (50 m) into the equation to find the maximum speed. The final calculation will yield the maximum speed v.

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Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the circle. For a car at the top of a hill, this force is provided by the gravitational force acting on the car. If the car's speed exceeds a certain limit, the gravitational force may not be sufficient to provide the necessary centripetal force, leading to the risk of the car losing contact with the road.
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Intro to Centripetal Forces

Gravitational Force

Gravitational force is the attractive force between two masses, such as the Earth and the car. At the top of the hill, this force acts downward and is crucial for providing the centripetal force needed to keep the car on its circular path. The gravitational force can be calculated using the formula F = mg, where m is the mass of the car and g is the acceleration due to gravity, approximately 9.81 m/s².
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Gravitational Forces in 2D

Maximum Speed at the Top of a Hill

The maximum speed at the top of a hill can be determined by balancing the gravitational force and the required centripetal force. At the critical point, the gravitational force equals the centripetal force needed to keep the car on the road. This relationship can be expressed mathematically as mg = mv²/r, where v is the speed and r is the radius of the hill. Solving for v gives the maximum speed the car can have without losing contact.
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Sphere on rough and smooth hills
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