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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
All textbooksYoung & Freedman Calc 14th EditionCh 05: Applying Newton's LawsProblem 8c
Chapter 5, Problem 8c

A 11301130-kg car is held in place by a light cable on a very smooth (frictionless) ramp (Fig. E5.85.8). The cable makes an angle of 31.0°31.0° above the surface of the ramp, and the ramp itself rises at 25.0°25.0° above the horizontal. How hard does the surface of the ramp push on the car?
An orange car on a frictionless ramp, held by a cable at a 31° angle, ramp inclined at 25°.

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Step 1: Begin by identifying the forces acting on the car. These include the gravitational force (weight), the tension in the cable, and the normal force exerted by the ramp. The gravitational force acts vertically downward, the tension in the cable acts along the direction of the cable, and the normal force acts perpendicular to the surface of the ramp.
Step 2: Break the gravitational force into components relative to the ramp. The component parallel to the ramp is given by \( F_{g, \text{parallel}} = m g \sin(\theta_{\text{ramp}}) \), and the component perpendicular to the ramp is \( F_{g, \text{perpendicular}} = m g \cos(\theta_{\text{ramp}}) \), where \( m \) is the mass of the car, \( g \) is the acceleration due to gravity, and \( \theta_{\text{ramp}} \) is the angle of the ramp above the horizontal.
Step 3: Analyze the forces perpendicular to the ramp. Since the car is stationary, the normal force \( F_{\text{normal}} \) must balance the perpendicular component of the gravitational force and the perpendicular component of the tension in the cable. The perpendicular component of the tension is given by \( T \sin(\theta_{\text{cable}}) \), where \( \theta_{\text{cable}} \) is the angle the cable makes with the ramp.
Step 4: Write the equation for the forces perpendicular to the ramp: \( F_{\text{normal}} = F_{g, \text{perpendicular}} - T \sin(\theta_{\text{cable}}) \). Substitute \( F_{g, \text{perpendicular}} = m g \cos(\theta_{\text{ramp}}) \) into the equation.
Step 5: Solve for \( F_{\text{normal}} \) using the given values: \( m = 1130 \ \text{kg} \), \( g = 9.8 \ \text{m/s}^2 \), \( \theta_{\text{ramp}} = 25.0^\circ \), and \( \theta_{\text{cable}} = 31.0^\circ \). You will also need to calculate the tension \( T \) from the forces parallel to the ramp, which can be done using a separate equation: \( T \cos(\theta_{\text{cable}}) = F_{g, \text{parallel}} \).

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Key Concepts

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

Forces on an Inclined Plane

When an object is on an inclined plane, several forces act on it, including gravitational force, normal force, and tension. The gravitational force can be resolved into components parallel and perpendicular to the surface of the ramp. Understanding how these forces interact is crucial for analyzing the motion and equilibrium of the car on the ramp.
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Normal Force

The normal force is the force exerted by a surface to support the weight of an object resting on it, acting perpendicular to the surface. In this scenario, the normal force counteracts the component of the gravitational force acting perpendicular to the ramp. Calculating the normal force is essential to determine how hard the ramp pushes on the car.
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Trigonometric Functions in Physics

Trigonometric functions, such as sine and cosine, are used to resolve forces into their components, especially in problems involving angles. In this case, the angles of the ramp and the cable allow us to calculate the components of the gravitational force and the tension in the cable. Mastery of these functions is vital for solving problems involving inclined planes and forces.
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Related Practice
Textbook Question

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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Textbook Question

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 floor.

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Textbook Question

Find the tension in each cord in Fig. E5.75.7 if the weight of the suspended object is ww.

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Textbook Question

An astronaut is inside a 2.25×1062.25 × 10^6 kg rocket that is blasting off vertically from the launch pad. You want this rocket to reach the speed of sound (331331 m/s) as quickly as possible, but astronauts are in danger of blacking out at an acceleration greater than 4g4g. 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 11301130-kg car is held in place by a light cable on a very smooth (frictionless) ramp (Fig. E5.85.8). The cable makes an angle of 31.0°31.0° above the surface of the ramp, and the ramp itself rises at 25.0°25.0° above the horizontal. Find the tension in the cable.

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Textbook Question

A 11301130-kg car is held in place by a light cable on a very smooth (frictionless) ramp (Fig. E5.85.8). The cable makes an angle of 31.0°31.0° above the surface of the ramp, and the ramp itself rises at 25.0°25.0° above the horizontal. Draw a free-body diagram for the car.

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views