Il The 1.0 kg physics book in FIGURE P7.41 is connected by a...

Key Concepts

Newton's Second Law

This principle states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. In problems involving motion along an inclined plane with friction and connected bodies, correctly applying Newton’s Second Law is essential to determine the acceleration of the system, the forces involved, and ultimately the distance the object travels before coming to a stop.

Inclined Plane Dynamics

This concept involves analyzing how forces act on objects placed on slopes. The gravitational force is decomposed into components parallel and perpendicular to the plane, and these components determine both the acceleration along the plane and the normal force. This framework is crucial when calculating the net force that influences motion and the frictional force acting against it.

Friction (Static and Kinetic)

Friction is the resistive force that opposes relative motion between surfaces in contact. Static friction prevents motion up to its maximum threshold, while kinetic friction comes into play once movement has begun. In this context, understanding both types of friction is key to predicting whether an object on an incline will remain stationary at the peak of its motion or slide back down, as well as determining its deceleration while moving.

Work-Energy Principle

The work-energy principle relates the work done by forces (including friction and gravitational components) to the change in kinetic energy of an object. This approach allows for the calculation of how far an object will travel on an incline by considering the energy lost to friction and gravity as the object’s kinetic energy is converted, thus providing an alternative method to Newton’s laws for solving motion problems.

Connected Systems and Tension

Analyzing systems where objects are connected by a string requires an understanding of how tension transmits forces between bodies. In such systems, the acceleration of one object is directly linked to that of the other. The interplay of gravitational forces, friction, and tension must be considered collectively to accurately predict the motion and equilibrium conditions of all parts of the system.

Transcript

00:01 Hi, here in this given problem there is a rough inclined plane making an angle of 20 degree with the horizontal like this.

00:19 The book kept over this inclined plane here this is an ideal pulley the cord passing over the pulley attached with the book and then finally there is a mug coffee mug attached at the other end of the cord.

00:50 Mass of the mug is small m so its weight mg acting vertically downward tension t in the string upward and here this is the same tension t over here also.

01:02 So, if the mass of the book is capital m its weight mg acting vertically downward so the two components of this weight one of the component normal to the inclined plane mg cos 20 degree and another component down the inclined plane mg sin 20 degree normal of the inclined plane at the book that is n.

01:35 So, when the book will be pushed upward over the inclined force of friction will be acting over it in downward direction kinetic friction.

01:48 Now the given data mass of the book 1 .0 kg mass of the coffee mug 500 g or we can say 0 .500 kg coefficient of static friction 0 .50 coefficient of kinetic friction 0 .20.

02:19 In the first part of the problem we have to find the distance moved by the book up along the inclined plane before coming to rest.

02:30 It is given an initial velocity vi is equal to 3 .0 m per second in upward direction.

02:39 Finally it comes to rest.

02:42 Now to find the distance we should find acceleration in its motion for which using free body diagram of the mug as the book will be moving up but the motion is retarded means downward force should be more.

02:57 Similarly here also the downward force means mg acting over the mug that should be more than t.

03:04 So, for the free body diagram of the mug mg minus t that should be equal to m into a.

03:23 We can make it equation number 1 and then taking free body diagram of the book total downward force acting on it tension t in the string plus component of its weight mg sin 20 degree plus force of kinetic friction mu k times normal reaction n which is equal to mg cos 20 degree and that should be equal to m into a that is equation number 2...

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