A 25.0-kg box of textbooks rests on a loading ramp that makes an angle αwith the horizontal. The

A 25.0-kg box of textbooks rests on a loading ramp that...

A $25.0-\mathrm{kg}$ box of textbooks rests on a loading ramp that makes an angle $\alpha$ with the horizontal. The coefficient of kinetic friction is $0.25,$ and the coefficient of static friction is $0.35 .$ (a) As the angle $\alpha$ is increased, find the minimum angle at which the box starts to slip. (b) At this angle, find the acceleration once the box has begun to move. (c) At this angle, how fast will the box be moving after it has slid 5.0 $\mathrm{m}$ along the loading ramp?

Key Concepts

Static Friction

Static friction is the force that resists the start of sliding motion between two surfaces. It acts parallel to the contact surface and has an upper limit, which is determined by the static friction coefficient multiplied by the normal force. This concept is crucial for determining when the object on an inclined plane will overcome the maximum static friction and begin to slide.

Kinetic Friction

Kinetic friction is the force that opposes the relative motion of surfaces sliding against each other once motion has begun. It usually has a constant value determined by the kinetic friction coefficient and the normal force. In problems involving motion along an inclined plane after slipping starts, kinetic friction is used to calculate the net force and the acceleration of the object.

Inclined Plane Dynamics

Inclined plane dynamics involves analyzing the forces acting on an object on a sloping surface, where gravity is decomposed into components parallel and perpendicular to the plane. This analysis is essential for understanding the normal force, frictional forces, and the net force that causes acceleration down the incline.

Newton's Second Law

Newton's Second Law states that the net force acting on an object is equal to the product of its mass and its acceleration. This principle is applied on an inclined plane to relate the gravitational force components, frictional forces, and the resulting acceleration of the object along the ramp.

Kinematics

Kinematics deals with the motion of objects without considering the forces that cause the motion. In the context of an object sliding along an incline, kinematic equations are used to compute properties such as the distance traveled, speed, and time of travel once the acceleration is known.

Transcript

00:01 And this problem, the key is knowing how to draw a free body diagram so we can see the forces acting on the object.

00:08 So if you think of a person sitting on a chair, they'll have a force due to gravity pointing straight down, and that's its weight.

00:15 So you have mass times gravity.

00:18 And acting in the exact opposite direction, you have the normal force.

00:22 And if the person on the chair is being pushed to the right, we'll have a force to the right.

00:28 And if it experiences friction, friction will be in the opposite direction of the motion.

00:35 This would be kinetic friction because there's movement.

00:40 So once you can draw that and see the forces, you can easily solve for either variable to get a number.

00:47 So in here we have a book that weighs 25 kilograms that's on a loading ramp.

00:56 And it experiences two types of friction.

00:58 First, kinetic friction.

00:59 So we're given the coefficient of kinetic friction.

01:04 This is friction due to movement, which is 0 .25.

01:07 And also static friction.

01:09 This is friction when it's just sitting there.

01:12 And we're given the coefficient of static friction, which is 0 .35.

01:19 And so for part a, we want to find the angle theta that will cause the book to slide down the ramp.

01:30 So it'll look like something like this, or the book.

01:34 Will begin to move down the ramp.

01:39 So let's first draw a free body diagram this book.

01:43 So again, we have the weight straight down.

01:47 And because it's at an angle, it'll have components.

01:50 So we have w .y in the y direction.

01:55 And then down the ramp will be weight in the x direction.

02:02 And the exact opposite of w .y will have the normal force.

02:09 And since it's about to slide down the ramp, it'll experience static friction in the opposite direction.

02:22 Static friction, the equation is coefficient of static friction times the normal force.

02:32 And for the weight, since it has components, we can say wx is w -sign theta, and w -y is w -cosined theta.

02:49 So first let's look at the forces in the x -rex.

02:53 Direction.

02:55 So looking back at a free body diagram, in the direction we have wx and we have static friction.

03:02 And if we say down the ramp and straight down is positive, that in the x direction we have wx, which is w sine theta minus static friction, which is mu s times the normal force, equals since it's about to move the acceleration is zero.

03:26 So we have for meta.

03:28 Maybe we have zero...

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