Two blocks of masses m1 and m2 are placed on a table in...

Two blocks of masses $m_{1}$ and $m_{2}$ are placed on a table in contact with each other as discussed in Example 5.7 and shown in Active Figure 5.12 a. The coefficient of kinetic friction between the block of mass $m_{1}$ and the table is $\mu_{1}$ , and that between the block of mass $m_{2}$ and the table is $\mu_{2} .$ A horizontal force of magnitude $F$ is applied to the block of mass $m_{1} \cdot$ We wish to find $P,$ the magnitude of the contact force between the blocks. (a) Draw diagrams showing the forces for each block. (b) What is the net force on the system of two blocks? (c) What is the net force acting on $m_{1}$ ? (d) What is the net force acting on $m_{2} ?$ (e) Write Newton's second law in the $x$ direction for each block. (f) Solve the two equations in two unknowns for the acceleration $a$ of the blocks in terms of the masses, the applied force $F,$ the coefficients of friction, and $g$ . (g) Find the magnitude $P$ of the contact force between the blocks in terms of the same quantities.

Key Concepts

Contact Force

The contact force refers to the force exerted by one object on another when they are in physical contact. In multi-body dynamics problems, understanding the role and calculation of contact forces is essential for establishing the internal interactions between bodies and ensuring consistency in applying Newton's laws to each individual mass.

System Analysis

System analysis involves treating multiple interacting objects as a single entity to simplify the force and motion analysis. By summing forces and masses, it is possible to determine the overall acceleration of the system before addressing the individual interactions, such as internal contact forces.

Net Force

The concept of net force involves the vector sum of all forces acting on an object or a system. It is the resultant force that determines the structure’s acceleration according to Newton’s second law, and it is calculated by considering all contributing forces including applied forces, frictional forces, and contact forces.

Newton's Second Law

Newton’s second law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This law (F = ma) is foundational for setting up and solving equations of motion in systems with multiple forces and objects.

Free-Body Diagram

A free-body diagram is a graphical illustration used to visualize all the forces acting on an individual object, separate from its surroundings. It helps in identifying forces such as gravity, normal force, friction, applied force, and any contact forces, which is essential for systematically solving problems using Newton’s laws.

Frictional Forces

Frictional forces oppose the motion of objects relative to each other or a contact surface. In problems involving kinetic friction, the force is calculated as the product of the coefficient of kinetic friction and the normal force. Understanding friction is crucial for accurately determining net forces and accelerations in dynamics problems.

Transcript

00:01 Now in this question we are looking at a similar example to example 7, right, so 5 .7, in which we have a, we have two blocks of mass resting on the table and we have an external force acting from the left and pushing the blocks together.

00:25 Now they have this, now the blocks are in contact with each other, so we have a contact force p.

00:33 Now the only difference between this and 5 .7 is that there is friction that we have to account for.

00:39 And that there's two different kind of two different coefficients of friction depending on the block.

00:46 So m1 has coefficient of kinetic friction of mu1, whereas for m2 is mu2.

00:54 Now we want to draw first of all the individual diagrams for each of the blocks.

01:00 So let us start with block 1 m1.

01:08 So what it experiences is the external force, of course, coming from the left.

01:14 It also experiences the frictional force because we assume that it's going to move towards the right, right? so there's frictional force opposing its motion, f -r, and we have gravity, then we have the normal force.

01:34 Finally we have the contact force.

01:42 So the contact force from m2 will be applying towards m1 from the right hand side towards the left.

01:55 So a more accurate representation would be like this because it is acting from the edge, right, from the right edge of the block.

02:11 Now on the other end for m2, a smaller block.

02:14 Again experiences friction as well and from the left -hand side you will experience the pushing force from our m1 basically is the contact force b and also the mass i mean the weight and the normal force the usual forces so this for m2 now to find the net force on the system of two blocks.

03:00 So we are considering the entire block here as a single system.

03:10 So what are the net forces acting on the this system will be for external force f as well as the fictional forces from me 1 and mu 2.

03:24 Now we know that the frictional forces is actually related to the coefficient.

03:30 By the normal force.

03:35 So the total friction we have mu1 and the normal force from the m1 is just equals to the weight right to balance out all the forces in the wide direction...

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