Ch 08: Dynamics II: Motion in a Plane

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 08: Dynamics II: Motion in a Plane

Problem 18

Chapter 8, Problem 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?

Verified step by step guidance

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:

F

_c = 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:

m g = m v

²/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 = v

²/r. Rearrange to solve for

v

v = \sqrt{g r}

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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Related Practice

Textbook Question

The normal force equals the magnitude of the gravitational force as a roller-coaster car crosses the top of a 40-m-diameter loop-the-loop. What is the car's speed at the top?

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Textbook Question

Communications satellites are placed in circular orbits where they stay directly over a fixed point on the equator as the Earth rotates. These are called geosynchronous orbits. The altitude of a geosynchronous orbit is 3.58 x 10⁷ m (approximately 22,00 miles). Astronomical data are inside the back cover of the book. What is the weight of a 2000 kg satellite in a geosynchronous orbit?

Textbook Question

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A 500 g ball moves in a vertical circle on a 102-cm-long string. If the speed at the top is 4.0 m/s, then the speed at the bottom will be 7.5 m/s. (You'll learn how to show this in Chapter 10.) What is the gravitational force acting on the ball?

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A satellite orbiting the moon very near the surface has a period of 110 min. What is free-fall acceleration on the surface of the moon? Astronomical data are inside the back cover of the book.

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Textbook Question

The weight of passengers on a roller coaster increases by 50% as the car goes through a dip with a 30 m radius of curvature. What is the car's speed at the bottom of the dip?

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