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 ($\textbf{Fig. E5.8}$). 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?Figure e5.8(Figure Cant copy)

Key Concepts

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.

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.

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.

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.

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.

Transcript

00:01 All right, in this problem, we have a car resting on a ramp being held up by a cable, which is inverted to the, is at an angle to the plane of the ramp.

00:19 So if that's parallel with the plane of the ramp, the dotted line there, we have this one at a 31 degree angle.

00:25 I'm going to label my block here at m.

00:28 That is my lovely drawn car.

00:29 And finally, the ramp is inclined at 25 degrees.

00:40 So in this case, we're asked to draw a free body diagram for the car in part a of this problem.

00:52 So first, in our free body diagram of the car, let's try the gravitational force.

00:59 We always know that is going to be there.

01:04 And if you are resting on some object, that object is pushing back up on you.

01:09 So there will be a normal force here extending out.

01:17 And this normal force will be at an angle of 25 degrees with the axis established by our gravitational force.

01:29 And then we will have a tension force due to the cable.

01:44 And if we think of this dotted line as perpendicular to normal force, then this is at 31 degrees.

01:58 So the intuitive referencing for me here is sticking along the parallel of the ramp and perpendicular to it.

02:09 So i'm going to, or actually, yeah, i'm going to convert both my tension force here and my gravitational force, which is also at 25 degrees with respect to the normal force here.

02:35 And i'm going to align it along this reference frame.

02:44 So let's redraw our free body diagram with all of our forces broken down into components...

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