∙∙A 1130-kg car is held in place by a light cable on a very smooth (frictionless) ramp, as shown

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

Key Concepts

Free-Body Diagram

A free?body diagram is a simplified representation of a system or object showing all the forces acting upon it, without any distraction from its environment. This diagram is crucial for visualizing how different forces, such as gravity, tension, normal force, and friction, interact and combine to produce equilibrium or acceleration.

Components of Forces on an Inclined Plane

When analyzing forces on an inclined plane, it is important to resolve forces into components that are parallel and perpendicular to the surface of the incline. This technique simplifies the problem by decoupling the effects of gravity and other forces along the directions that are most relevant to the motion or equilibrium of the object.

Equilibrium Conditions

In problems where an object is stationary or moving at constant velocity, the equilibrium conditions state that the net force in any direction must equal zero. By setting up equations based on the balance of forces along each axis, one can solve for unknown forces such as tension or the normal force.

Normal Force

The normal force is the reactive force exerted by a surface that prevents an object from penetrating it. In the context of inclined planes, the normal force acts perpendicular to the surface and is a key component when resolving the object’s weight into perpendicular and parallel components.

Tension Force

The tension force in a cable or rope is the pulling force transmitted along its length. When used to hold an object in place or to accelerate it, the tension force must be carefully analyzed in terms of its magnitude and direction, particularly by resolving any angled forces along the appropriate axes.

Transcript

00:01 All right, a car that is on an incline is being pulled up by a cable at 31 degrees from 30 that makes a 31 degree angle with the incline.

00:14 Okay, the incline itself is at 25 degrees.

00:18 Okay, so let's start with the free body diagram, part a.

00:22 And so you have, obviously, your force of gravity, weight acting downwards and then you have a certain so that's acting downwards then you have the tension from the cable acting upwards and this is at an angle of 31 degrees from the incline so there's an angle of 25 degrees here because that's the angle that's how inclined the ramp is 25 degree angle between the vertical axis that we just know we're setting up.

01:12 Let me draw that a little better.

01:19 But essentially an angle, essentially the vertical axis should be that axis.

01:38 And so the normal force is acting on the car from the, by the ramp and so that's acting directly that's acting vertically upwards and okay so the the convention we chose here is is that the x and y axes are sort of like that not exactly but that's where there are so they're inclined okay they're not the traditional x and y like this that we're used to.

02:16 Anyway, so that so this free body diagram has the tension at 31 degrees from the incline from the x -axis right and then normal force directly vertically and then force of gravity is 25 degrees below the or at an angle of 25 degrees from the vertical axis.

02:43 Okay so this is the free body slightly complicated but note that changing your axes to the usual stuff is fine as long as to the usual this crossed out way is fine you just have to make sure your angles are right in your calculations and so part b you want to find tension on the cable and so for this you of course use newton's second law resolve forces in the, let's look at the x direction.

03:23 For the x direction, there's equilibrium, same for the y direction.

03:30 So that force is zero.

03:33 And so what you have is the x component of tension added to the y component, to the x component of gravity to be equal to zero.

03:46 Okay, so x component of tension is just t.

03:49 So t is acting that way and you have a 31 degree angle.

03:55 Your x component would be this.

03:57 So t so you need a cosine there t coast 31 right and then your weight which is acting this where and you have downwards and you have an angle of 25 degrees there.

04:15 So um so basically this um so basically you have w acting this way.

04:27 This is the direction you want along your x -axis.

04:34 So for that, you would need a sign there because this is really what you want...

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