A block of mass m1=21.9 kg is at rest on a plane inclined at θ=30.0^∘ above the horizontal. The

A block of mass m1=21.9 kg is at rest on a plane inclined...

A block of mass $m_{1}=21.9 \mathrm{~kg}$ is at rest on a plane inclined at $\theta=30.0^{\circ}$ above the horizontal. The block is connected via a rope and massless pulley system to another block of mass $m_{2}=25.1 \mathrm{~kg}$, as shown in the figure. The coefficients of static and kinetic friction between block 1 and the inclined plane are $\mu_{s}=0.109$ and $\mu_{\mathrm{k}}=0.086,$ respectively. If the blocks are released from rest. what is the displacement of block 2 in the vertical direction after 1.51 s? Use positive numbers for the upward direction and negative numbers for the downward direction.

Key Concepts

Newton's Second Law for Connected Systems

This concept involves applying Newton's Second Law (F = ma) to each component of a system that is interconnected, such as blocks linked by a rope over a pulley. In these systems, the acceleration of each mass is connected by the constraints imposed by the rope, and solving the resulting equations allows one to determine the common acceleration and the internal forces (such as tension).

Friction on Inclined Planes

When an object is on an inclined surface, friction plays a crucial role in its motion. The frictional force, which opposes motion, is calculated by multiplying the coefficient of friction (static or kinetic) with the normal force. The normal force on an inclined plane is reduced compared to a horizontal surface because it is a component of the weight of the object, determined using the cosine of the inclination angle.

Free-Body Diagram Analysis

Free-body diagrams are graphical representations that illustrate all forces acting on each object in a system. They help in systematically identifying and setting up the forces, such as gravitational, frictional, normal, and tension forces, which then can be used to apply Newton's laws to analyze the motion of objects.

Kinematics Equations for Constant Acceleration

Kinematics equations describe the motion of an object under constant acceleration. These equations relate displacement, velocity, acceleration, and time, thus allowing one to calculate how far an object moves over a certain period given its acceleration. This is essential when the acceleration of the system is determined from the dynamics and then used to find displacement.

Mechanical Constraint in Pulley Systems

In pulley systems with ropes, the motion of connected objects is often constrained such that they share the same magnitude of acceleration. This constraint arises because the rope is typically inextensible, meaning that as one object moves, the other must move correspondingly. Understanding these constraints is critical in linking the individual equations of motion to solve for system behavior.

Transcript

00:02 Hi, this is the problem of connected motion in which there is an inclined plane at which a block has been kept.

00:19 This is a rough inclined plane.

00:23 Mass of the block kept over it is m1.

00:26 Then there is a frictionless massless pulley, a thread passing over it.

00:35 And another block of mass m2 is attached with the other end of this row.

00:45 Now weight of this hanging block, m2g, it will be bringing this block down.

00:53 It will create a tension t in this row and the tension t will be same here also.

01:01 Angle theta, that angle theta is given as 30 .0.

01:07 Degree, mass m1 is 21 .9 kilogram, m2 is 25 .1 kilogram and coefficient of kinetic friction we will use here, that is 0 .086 because the block will be in motion.

01:29 Now the mass, the weight of the block, this m1, which is over the inclined surface, acting vertically downward, m1g, its component, because if this angle is theta, here this angle will also be theta.

01:45 The component of this weight perpendicular to the incline plane, that is m1g cross theta, and along the inclined plane downward, that is m1g sine theta, as if the block m2 is moving down, suppose with an acceleration a, this block m1 will be moving up with the same acceleration hence a force of friction will be acting opposite to the direction of motion and that force of friction is given by the expression mu k into n where n is the normal reaction applied by the surface inclined surface of the over the block and that n will be equal to m1g cos theta.

02:35 So this force of friction will be given as mu k into m1g cos theta...

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