A 1130 -kg car is held in place by a light cable on a very...

A 1130 -kg car is held in place by a light cable on a very smooth (frictionless) ramp, as shown in Fig. E5.8. The cable makes an angle of $31.0^{\circ}$ above the surface of the ramp, and the ramp itself rises at $25.0^{\circ}$ above the horizontal. (a) Draw a free-body diagram for the car. (b) Find the tension in the cable. (c) How hard does the surface of the ramp push on the car?

Key Concepts

Normal Force

The normal force is the perpendicular contact force exerted by a surface on an object resting upon it. On an inclined plane, this force is not equal to the gravitational force but is adjusted based on the angle of the inclination. It plays a key role in ensuring that objects do not penetrate or lift off the surface, and is essential in analyzing contact problems.

Tension Force

The tension force is the pulling force transmitted through a cable, rope, or string when it is subjected to a force. In contexts involving inclined planes and cables, calculating the tension involves balancing the force components along the line of the cable with those due to other forces such as gravity.

Force Resolution on Inclined Planes

Force resolution involves breaking forces into components that are parallel and perpendicular to the inclined surface. This technique simplifies calculations by aligning the components with the directions of motion and constraints. It is especially important in problems with inclined planes, as many forces do not act directly along the standard horizontal and vertical axes.

Free-Body Diagram

A free-body diagram is a sketch that shows all the forces acting on an object, represented as vectors. It is used in mechanics to help visualize the influence of forces like gravity, tension, normal force, and friction on an object. In problems involving inclined planes, it is crucial to represent these forces correctly and choose an appropriate coordinate system to simplify analysis.

Static Equilibrium

Static equilibrium refers to a state where an object remains at rest because all the forces acting on it balance out. In such systems, the sum of forces in all directions equals zero. Uniformly applying equilibrium conditions is vital for solving problems where forces like tension and normal force are unknown.

Transcript

00:01 Okay, so in the problem 8, we have, let me see, we have a car held in place by a cable.

00:13 So let's draw here the problem.

00:16 We have a ramp with an inclination of 25 degrees.

00:25 We have a car above this ramp being held in place by a cable, which has a, angle with the plan of 31 degrees, okay? and this is the sketch of the problem.

00:48 And the first thing we need to answer in this problem is the free body diagram.

00:55 So we need to sketch the free body diagram.

00:59 Then we need to calculate what is the tension in the cable.

01:03 And finally, how hard the surface on the ramp off the ramp pushes the car.

01:10 So basically what we need.

01:12 Let's represent the free body diagram first.

01:16 So imagine that we have the body here and the sketch of the free body diagram will be, first of all, we have the normal force pointing up.

01:31 So this is the normal force.

01:33 This is the plane and we have a cable.

01:39 With a tension here making an angle of 31 degrees.

01:45 This is tension t.

01:47 And finally we have the weight.

01:50 So the weight of the body, let's call this p, is going to be with an angle of 25 degrees.

02:02 Okay, so this is the free body diagram, all the forces that exert influence on this object on this car.

02:14 So what we want in the first item, we want the tension.

02:18 So in the item b, we want the tension, and on item c, we want the normal...

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