Figure (6-E12) shows a small block of mass m kept at the left end of a larger block of mass M an

Figure (6-E12) shows a small block of mass m kept at the...

Key Concepts

Newton's Second Law of Motion

Newton's Second Law states that the net force acting on an object is equal to the object's mass multiplied by its acceleration. This principle is fundamental for analyzing the motion of each body in multi-body systems, enabling us to determine how forces such as friction affect acceleration and displacement.

Friction

Friction is the resistive force that arises when two surfaces interact, opposing their relative motion. It is quantified by the friction coefficient and plays a vital role in dynamics by influencing the net forces and accelerations experienced by objects in contact with different surfaces.

Relative Motion

Relative motion involves analyzing the movement of one object with respect to another. This concept is crucial in systems where multiple bodies interact, as it helps determine when one object moves away from or separates from another due to differences in acceleration or velocity.

Dynamics of Interacting Bodies

In systems with interacting bodies, the dynamics are governed by the combined effects of forces acting on each object. Analyzing such systems typically involves setting up and solving coupled equations of motion that account for the individual forces, friction, and constraints affecting the movement of each body.

Constraint Conditions

Constraint conditions define the restrictions on the relative motions of bodies that are in contact or connected. They help establish relationships between the displacements, velocities, or accelerations of different parts of a system, and are essential for determining specific events, such as the point of separation between two bodies.

Transcript

00:01 In this problem, we have two blocks of different masses on top of each other.

00:07 The blocks are moving at initial velocity v and accelerating on this direction.

00:18 Now, the problem asks us what would be the time after the smaller block separate with the large block, given that the length of the large block is l.

00:35 The coefficient of friction between the road and the large block is also given, which is mu, and the coefficient of friction between the two blocks is mu over two.

00:51 So this problem involves kinematic equations and newton's law of motion.

00:56 To solve this problem, we must first derive an expression for the acceleration.

01:02 We can achieve that by using the force equations of each blocks.

01:07 Let's start first with the small block.

01:11 Assuming this is the block, the forces acting on its body are its weight, the normal force, and on this direction is the friction.

01:29 Using first law, the summation of forces on the vertical plane is equal to zero since the body is in equilibrium on this plane.

01:40 Its movement is horizontal and not vertically.

01:45 Then the forces involved in this plane are normal force and the weight.

01:54 We denote this as negative since it is opposite the normal force.

02:00 And we know that the weight of the block is just equal to its mass multiplied by the acceleration due to gravity.

02:11 Now for the horizontal plane, the summation of the force is not equal to zero.

02:22 Since the block is accelerating.

02:25 We will now use second law of motion.

02:28 This states that the net force is equals to the product of the mass of the object and its acceleration due to its motion.

02:41 In this case, the horizontal force is the friction.

02:48 We denote this as negative since it is opposite the direction of the acceleration.

02:55 Since the black is moving, the friction force is just equal to the product of the coefficient of friction given, which is mu over 2, and the normal force.

03:11 We know that normal force is the same magnitude with the weight.

03:16 As such, we can express this equation like this.

03:24 Simplifying, then we have a1 is equal to negative mu over 2.

03:37 Now let's move to the larger block.

03:43 Assuming this is the larger block, for the horizontal plane, the forces acting on it are the friction force between the road and on the right is the friction force between the small block.

04:16 Pointed upwards is the normal force of the large block, downwards is its weight, and another also downward is the normal force of the smaller block since this is under the smaller block.

04:39 Again, summation of this four, the vertical forces is equal to zero since the movement is horizontal.

04:51 Then the forces on this plane are the normal force of the large block, the normal force of the smaller block, and the weight of the large block...

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