Ch 05: Applying Newton's Laws

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 05: Applying Newton's Laws

Problem 9a

Chapter 5, Problem 9a

A man pushes on a piano with mass $180$ kg; it slides at constant velocity down a ramp that is inclined at $19.0 degrees$ above the horizontal floor. Neglect any friction acting on the piano. Calculate the magnitude of the force applied by the man if he pushes parallel to the incline.

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Identify the forces acting on the piano. Since the piano is sliding at constant velocity, the net force along the incline must be zero. This means the force applied by the man balances the component of the piano's weight acting along the incline.

Write the expression for the gravitational force acting on the piano. The weight of the piano is given by $ F_g = m \cdot g $, where $ m = 180 \; \text{kg} $ is the mass of the piano and $ g = 9.8 \; \text{m/s}^2 $ is the acceleration due to gravity.

Determine the component of the gravitational force acting along the incline. This is given by $ F_{g, \text{parallel}} = F_g \cdot \sin(\theta) $, where $ \theta = 19.0^\circ $ is the angle of the incline.

Since the piano is moving at constant velocity, the force applied by the man, $ F_{\text{man}} $, must exactly counteract $ F_{g, \text{parallel}} $. Therefore, $ F_{\text{man}} = F_g \cdot \sin(\theta) $.

Substitute the known values into the equation $ F_{\text{man}} = m \cdot g \cdot \sin(\theta) $ to calculate the magnitude of the force applied by the man. Ensure the angle $ \theta $ is converted to radians if necessary for trigonometric calculations.

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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. In this scenario, since the piano slides at constant velocity, the net force acting on it is zero, meaning the applied force by the man must balance the component of gravitational force acting down the ramp.

Gravitational Force Component

When an object is on an inclined plane, the gravitational force acting on it can be resolved into two components: one parallel to the incline and one perpendicular to it. The parallel component, which causes the object to slide down the ramp, is calculated using the formula F_parallel = mg sin(θ), where m is the mass, g is the acceleration due to gravity, and θ is the angle of the incline.

Constant Velocity

Constant velocity implies that an object's speed and direction remain unchanged over time. In this case, since the piano is moving down the ramp at constant velocity, the forces acting on it are balanced. This means the force exerted by the man must equal the gravitational force component acting parallel to the incline, allowing for a straightforward calculation of the required force.

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Related Practice

Textbook Question

A man pushes on a piano with mass $180$ kg; it slides at constant velocity down a ramp that is inclined at $19.0°$ above the horizontal floor. Neglect any friction acting on the piano. Calculate the magnitude of the force applied by the man if he pushes parallel to the floor.

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Textbook Question

An astronaut is inside a $2.25 \times 10^{6}$ kg rocket that is blasting off vertically from the launch pad. You want this rocket to reach the speed of sound ( $331$ m/s) as quickly as possible, but astronauts are in danger of blacking out at an acceleration greater than $4 g$ . What is the maximum initial thrust this rocket's engines can have but just barely avoid blackout? Start with a free-body diagram of the rocket.

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Textbook Question

A $1130$ -kg car is held in place by a light cable on a very smooth (frictionless) ramp (Fig. E $5.8$ ). The cable makes an angle of $31.0°$ above the surface of the ramp, and the ramp itself rises at $25.0°$ above the horizontal. How hard does the surface of the ramp push on the car?

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Textbook Question

A $1130$ -kg car is held in place by a light cable on a very smooth (frictionless) ramp (Fig. E $5.8$ ). The cable makes an angle of $31.0 degrees$ above the surface of the ramp, and the ramp itself rises at $25.0 degrees$ above the horizontal. Find the tension in the cable.

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Textbook Question

An astronaut is inside a $2.25 × 10^6$ kg rocket that is blasting off vertically from the launch pad. You want this rocket to reach the speed of sound ( $331$ m/s) as quickly as possible, but astronauts are in danger of blacking out at an acceleration greater than $4g$ . What force, in terms of the astronaut's weight $w$ , does the rocket exert on her? Start with a free-body diagram of the astronaut.

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Textbook Question

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A man pushes on a piano with mass 180180 kg; it slides at constant velocity down a ramp that is inclined at 19.0°19.0° above the horizontal floor. Neglect any friction acting on the piano. Calculate the magnitude of the force applied by the man if he pushes parallel to the incline.

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Key Concepts

Newton's Second Law of Motion

Gravitational Force Component

Constant Velocity

A man pushes on a piano with mass $180$ kg; it slides at constant velocity down a ramp that is inclined at $19.0 degrees$ above the horizontal floor. Neglect any friction acting on the piano. Calculate the magnitude of the force applied by the man if he pushes parallel to the incline.