A 113011301130-kg car is held in place by a light cable on a very smooth (frictionless) ramp (Fig. E5.85.85.8). The cable makes an angle of 31.0°31.0°31.0° above the surface of the ramp, and the ramp itself rises at 25.0°25.0°25.0° above the horizontal. Draw a free-body diagram for the car.

Ch 05: Applying Newton's Laws

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 05: Applying Newton's Laws

Problem 8a

Chapter 5, Problem 8a

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. Draw a free-body diagram for the car.

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Start by identifying all the forces acting on the car. These include the gravitational force (weight of the car), the tension in the cable, and the normal force exerted by the ramp. The gravitational force acts vertically downward, the tension force acts along the cable, and the normal force acts perpendicular to the surface of the ramp.

Draw the free-body diagram. Represent the car as a point. Draw an arrow pointing downward to represent the gravitational force $ F_g = m g $, where $ m $ is the mass of the car and $ g $ is the acceleration due to gravity. Label this force $ F_g $.

Add the tension force $ T $ as an arrow pointing along the direction of the cable. Since the cable makes an angle of 31.0° above the ramp, the tension force will be at this angle relative to the ramp's surface. Label this force $ T $.

Add the normal force $ F_N $ as an arrow perpendicular to the surface of the ramp. This force is exerted by the ramp on the car and prevents the car from sinking into the ramp. Label this force $ F_N $.

Finally, resolve the gravitational force $ F_g $ into two components: one parallel to the ramp ($ F_{g, \parallel} = F_g \sin(25.0°) $) and one perpendicular to the ramp ($ F_{g, \perp} = F_g \cos(25.0°) $). These components will help analyze the forces along and perpendicular to the ramp in subsequent calculations.

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Key Concepts

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

Free-Body Diagram

A free-body diagram is a graphical representation used to visualize the forces acting on an object. It isolates the object from its surroundings and shows all the forces, including their directions and magnitudes. This tool is essential for analyzing the dynamics of the object, allowing one to apply Newton's laws of motion effectively.

Forces on an Incline

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 angle of the incline affects these components, which are crucial for understanding how the object interacts with the surface and any other forces, such as tension in the cable.

Tension in a Cable

Tension is the force transmitted through a cable or rope when it is pulled tight by forces acting from opposite ends. In this scenario, the tension in the cable counteracts the component of gravitational force acting down the ramp. Understanding how to calculate and represent tension is vital for solving problems involving objects in equilibrium on inclined surfaces.

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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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