Ch. 04 - Dynamics: Newton's Laws of Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 04 - Dynamics: Newton's Laws of Motion

Problem 40a

Chapter 4, Problem 40a

At the instant a race began, a 65-kg sprinter exerted a force of 720 N on the starting block at a 22° angle with respect to the ground. What was the horizontal acceleration of the sprinter?

Verified step by step guidance

Identify the forces acting on the sprinter. The sprinter exerts a force of 720 N at an angle of 22° with respect to the ground. This force can be broken into two components: a horizontal component and a vertical component.

Calculate the horizontal component of the force using the formula:

F

_h = F ⋅ cos(θ), where

F

is the total force (720 N) and

θ

is the angle (22°).

Determine the net horizontal force acting on the sprinter. Since the problem does not mention any opposing forces (like friction), assume the horizontal component of the applied force is the net force.

Use Newton's second law of motion to calculate the horizontal acceleration. The formula is:

a = F

_h / m, where

F

_h is the horizontal force and

m

is the mass of the sprinter (65 kg).

Substitute the values into the equations and simplify to find the horizontal acceleration. Ensure the units are consistent (N for force, kg for mass, and m/s² for acceleration).

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Key Concepts

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

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed by the formula F = ma, where F is the net force, m is the mass, and a is the acceleration. Understanding this law is crucial for calculating the horizontal acceleration of the sprinter based on the force exerted.

Components of Force

When a force is applied at an angle, it can be resolved into horizontal and vertical components using trigonometric functions. The horizontal component can be found using the cosine of the angle, while the vertical component uses the sine. In this case, the horizontal force component will be essential for determining the sprinter's horizontal acceleration.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It provides the framework for analyzing how objects move, including concepts like acceleration, velocity, and displacement. In this scenario, kinematic principles will help relate the calculated horizontal force to the resulting acceleration of the sprinter.

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Textbook Question

A 27-kg chandelier hangs from a ceiling on a vertical 3.4-m-long wire. What will be the tension in the wire?

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Textbook Question

At the instant a race began, a 65-kg sprinter exerted a force of 720 N on the starting block at a 22° angle with respect to the ground. If the force was exerted for 0.32 s, with what speed did the sprinter leave the starting block?

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Textbook Question

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A 27-kg chandelier hangs from a ceiling on a vertical 3.4-m-long wire. What horizontal force would be necessary to displace its position 0.15 m to one side?

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Textbook Question

Superman must stop a 120-km/h train in 150 m to keep it from hitting a stalled car on the tracks. If the train's mass is 3.6 x 10⁵ kg, how much force must he exert? Compare to the weight of the train (give as %). How much force does the train exert on Superman?

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