Name: solving a two-mass pulley momentum problem
Uploaded: 2024-03-07T13:22:20.000Z
Duration: 1 min 23 s
Channel: JoVE

K12 Standards

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NC K-12 Science Standard Course of Study

9-12

Physics

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Motion and Stability- Forces and Interactions

PS.Phy.1.2

principle-linear-impulse-momentum-for-single-particle-problem

Principle of Linear Impulse and Momentum for a Single Particle: Problem Solving

Consider a massless pulley system consisting of a wooden box and a cylinder of known masses. Determine the speed of the attached objects at a specific time after the system is released from rest.
For the analysis, establish an appropriate coordinate system to determine the length of the entire rope. The time derivative of this length equation gives the velocity equation.
Draw a free-body diagram for the cylinder, indicating its weight and the tension in the rope. Apply the principle of linear impulse and momentum to the cylinder.
Here, the initial momentum is zero, while the impulse is due to the weight of the cylinder and the tension on the rope.
Similarly, draw a free-body diagram for the wooden box, indicating all the forces acting on it. Apply the principle of impulse and momentum and substitute the known values to simplify the equation.
Now, solve the three equations simultaneously to obtain the unknown velocities.
Assuming the downward velocity as positive, the negative velocity of the wooden box indicates an upward motion of the box.

principle of linear impulse and momentum for a single particle: problem solving

Consider a wooden box and a cylinder of known masses m1 and m2, respectively, hanging from a ceiling with the help of a massless pulley system.
<img alt="Pulley system diagram with static equilibrium, showing masses m1 and m2 on ropes and pulleys." class="xfigimg" src="/files/ftp_upload/15525/15525_PT_Figure_001.svg" data-altupdatedat="1762173717000" data-alt="Figure 1" />
The system is initially at rest and then released. What will be the velocities of the wooden box and cylinder at a specific time after the system has been released from the rest?
Here, the entire length of the rope is expressed as the combination of smaller segments attached to the wooden box and cylinder. As the system moves, both the wooden box and cylinder attain some velocities, but the entire length of the string remains constant. Therefore, the velocity expression is derived by taking the time derivative of the length of the entire rope.
Then, a free-body diagram is drawn for the cylinder, showing all the forces acting on it. Here, the integral of the net force acting on the cylinder for a given time interval t, equals the change in momentum of the cylinder. Similarly, a free-body diagram is drawn for the wooden box, and a corresponding equation is written.
<img alt="Static equilibrium equation, m2g-T=m2vc/t, physics formula, diagram." src="/files/ftp_upload/15525/15525_Equation_PT_001.svg" style="text-align: center; height: 32px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
<img alt="Static equilibrium equation in mechanics; tension and weight forces balance; physics formula." src="/files/ftp_upload/15525/15525_Equation_PT_002.svg" style="text-align: center; height: 32px;" data-altupdatedat="1762173717000" data-alt="Equation 2" />
Solving the above two equations simultaneously gives the cylinder and wooden box velocities. Here, it is important to note that the directions of the velocities for the wooden box and cylinder are opposite.

Consider a massless pulley system consisting of a wooden box and a cylinder of known masses. Determine the speed of the attached objects at a specific ...

Education

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Mechanical Engineering

Unit 1

Kinetics of a Particle: Impulse and Momentum

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KEY TERMS AND CONCEPTS

solving a two-mass pulley momentum problem

A massless pulley system with a wooden box and a cylinder shows how linear impulse and momentum can be used to find motion after release. The two objects have known masses, m1 and m2, and the system starts from rest before it is released.

The rope length stays constant as the box and cylinder move. That means the rope can be written as a set of smaller segments attached to each object. Taking the time derivative of the full rope length gives the velocity relation between the wooden box and the cylinder.

Next, free-body diagrams are drawn for both objects. A free-body diagram is a sketch of the forces acting on an object. For the cylinder, the integral of the net force over the time interval t equals the change in momentum. A similar equation is written for the wooden box.

The two equations are solved together to find the velocities of both objects at the chosen time after release. The wooden box and cylinder move in opposite directions, so their velocity directions are opposite.

Solving a Two-Mass Pulley Momentum Problem

solving-a-two-mass-pulley-momentum-problem

principle of linear impulse and momentum for a single particle

principle-of-linear-impulse-and-momentum-for-a-single-particle

Momentum and Impulse Diagrams for a Particle

Linear momentum describes how a particle moves in physics. It is a vector quantity, so it has both size and direction. Its magnitude equals mass times velocity, and its direction matches the particle’s velocity. Linear impulse, also called momentum impulse, describes the change in linear momentum. It is also a vector quantity. Impulse equals force multiplied by the time interval over which the force acts. For a particle of mass m in an inertial frame, the equation of motion can be integrated...

A massless pulley system with a wooden box and a cylinder shows how linear impulse and momentum can be used to find motion after release. The two objects have known masses, m1 and m2, and the system starts from rest before it is released. The rope length stays constant as the box and cylinder move. That means the rope can be written as a set of smaller segments attached to each object. Taking the time derivative of the full rope length gives the velocity relation between the wooden box and the...

principle of linear impulse and momentum for a system of particles

principle-of-linear-impulse-and-momentum-for-a-system-of-particles

Center of Mass and Particle Momentum

A system of particles follows an equation of motion that describes how external forces affect its motion in an inertial frame of reference. The particles also exert internal forces on one another, but these forces act in equal and opposite collinear pairs. Because they cancel, they do not appear in the equation of motion. The principle of linear impulse and momentum comes from integrating that equation of motion over time. After the limits are applied, the combined initial linear momenta of...

conservation of linear momentum for a system of particles

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Momentum Transfer in Billiard Collisions

Momentum transfer in billiard collisions shows how linear momentum changes when one ball strikes another. A moving cue ball hits a stationary ball and gives up part of its linear momentum. The cue ball slows down, and the other ball starts moving. The collision force acts for an extremely short time, so its impulse is negligible. In impulse-momentum analysis, forces are grouped as impulsive or non-impulsive. Impulsive forces act briefly but can cause a large momentum change. For a system of...

impact

Impact, Impulse, and Restitution

Impact describes what happens when two bodies collide and impulsive forces act between them. In impact mechanics, the colliding particles move along the line of impact. This line passes through their centers and is perpendicular to the contact plane. When particles with different initial velocities collide, they apply equal and opposite impulses and cause deformation. At maximum deformation, the particles move together with the same velocity. After that, restitution begins. Restitution is the...

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Central and Oblique Collision Dynamics

Impact events are described by how two objects collide and how they move after the collision. The two main forms are central impact and oblique impact. A central impact happens when objects strike head-on, with opposite velocities along the line of impact. An oblique impact happens when objects collide at an angle. In this case, both direction and velocity can change after the collision. This makes the motion after impact more complex than in a head-on collision. The coefficient of...

impact: problem solving

Mars Projectile Rebound Height Calculation

Mars projectile rebound height calculation uses kinematics and the coefficient of restitution. A rover launches a projectile, and the projectile hits the Martian surface before bouncing upward again. The goal is to find the maximum height it reaches after the collision. The launch point is taken as the origin. Upward velocity is treated as positive, and the horizontal velocity stays constant. Using the known initial velocity and Mars gravity, the vertical velocity at impact is calculated with...

angular momentum

Angular Momentum and Direction

Angular momentum describes rotational motion and the direction of that motion. It is defined as the moment of a particle’s linear momentum about a chosen point O. This makes it useful for describing how motion curves around a point. For a particle moving along a curved path in the x-y plane, the scalar form gives the magnitude of angular momentum. It uses the moment arm, d, which is the perpendicular distance from point O to the line of action of the linear momentum. Even when written in...

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Torque and Angular Momentum in Rotation

Torque and angular momentum explain how force changes rotation. When a force acts at a distance from the center, it creates torque. A spinning top is a simple example of this idea. Moment is the rotational form of force. It shows how a force can start rotational motion. Angular momentum is the rotational form of linear momentum. It describes an object’s tendency to keep rotating. The rate of change of angular momentum can be written as a derivative. In that expression, the first term becomes...

principle of angular impulse and momentum

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Angular Impulse, Momentum, and Rotation

Angular impulse and momentum describe how forces change an object's rotation over time. The idea connects the moment of force, or torque, with angular momentum. When the equation is integrated from an initial time to a final time, it gives the angular impulse and momentum principle. The principle defines initial and final angular momentum as the moments of a particle's linear momentum at those specific times. Angular impulse is found by integrating the moments of all forces acting on the...

principle of angular impulse and momentum: problem solving

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Angular Impulse and Final Velocity

Angular impulse and momentum help determine the final velocity of a ball attached to a massless rod. The ball has mass m and moves under a time-dependent torque. If the initial velocity is known, the final velocity after a time interval can be found with this principle. The first step is to draw a free-body diagram. A free-body diagram shows all the forces acting on the system. It helps make the motion and rotation easier to analyze. Next, the principle of angular impulse and momentum is...

steady flow of a fluid stream

Fluid Force and Momentum in a Pipe

Fluid force and momentum in a pipe are linked through a control volume with solid boundaries. As fluid moves through the pipe, the walls can change its direction by exerting an impulse through the resulting force. This setup helps show how flow behaves inside a fixed region. For steady flow, the mass entering the control volume at time t with velocity v1 equals the mass leaving after a small time dt with velocity v2. Over that same interval, the momentum inside the control volume stays...

key terms and concepts

Use linear impulse and momentum to solve a massless pulley system with a wooden box and cylinder. Find their velocities after release from rest.

Solving a Two-Mass Pulley Momentum Problem | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

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Ohio’s Learning Standards and Model Curriculum

Forces, Momentum And Motion

Forces in two dimensions

Tennessee Academic Standards

High School

Motion and Stability: Forces and Interactions

PHYS.PS2.14

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PS.8

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Internal and External Forces

Newton&#39;s first law states that a net external force causes a change in motion. External forces act on an object or system, originating outside of the ...

Newton's Second Law

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Mass and Weight

Mass and weight are often used interchangeably in everyday conversation. For example,&#160; medical records often show our weight in kilograms, but never ...

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Carry out investigations to explain the interactions of forces on an object according to Newton's Laws of Motion.

PS.Phy.2.2

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Analyze the motion of objects using time, distance, displacement, speed, velocity, and acceleration.

Solving a Two-Mass Pulley Momentum Problem

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