Atwood's Machine and Motion on an Inclined Plane PDF

Atwood’s machine and Motion on an Inclined Plane Dr. Racha Cheaib  Atwood’s Machine Atwood’s machine consists of two objects of unequal mass that are suspended vertically by a string over a light frictionless pulley. If one of the two masses is greater than...

Atwood’s machine and Motion on an Inclined Plane Dr. Racha Cheaib  Atwood’s Machine Atwood’s machine consists of two objects of unequal mass that are suspended vertically by a string over a light frictionless pulley. If one of the two masses is greater than the other, the system accelerates. The best examples of Atwood’s machine are elevators, garage doors, or wells.  Atwood’s Machine In an Atwood’s Machine, the difference in weight between two hanging masses determines the net force acting on the system of both masses. The heavier mass accelerates downward, and the lighter mass accelerates upward. This acceleration is due to gravity.  Acceleration due to Gravity Acceleration due to gravity, is the acceleration experienced by an object due to the gravitational force exerted by a massive body, such as Earth. Motion of a Bungee jumper Motion of Sky divers  Acceleration due to Gravity On Earth, the standard value of acceleration due to gravity is approximately: 2 g = 9.81m/s This means that, in the absence of air resistance, an object in free fall near the Earth's surface will accelerate at a rate of 9.81 meters per second squared.  Acceleration due to Gravity If you throw a ball from a cliff, every second the speed of the ball increases by 9.8 m/s in the absence of air resistance. This change in speed is due to the force of gravity.  Acceleration due to Gravity Acceleration due to gravity is also called free fall. However, that does not mean that the object is only moving downwards. Objects thrown upward or downward and those released from rest are all the examples of free fall. The standard equations of a uniformly accelerated body are applicable for free fall: Acceleration due to Gravity Acceleration is constant as a function of time. Acceleration due to Gravity Velocity decreases by 9.8 m/s every second. The plot is linear. Acceleration due to Gravity The position change is not linear, it is a quadratic relationship. Newton’s Second Law  Atwood’s Machine Assuming that the pulley has no mass, the string does not stretch, and that there is no friction, we have: +a The acceleration of the system of both masses can be expressed as:  Sample Problem Determine the acceleration of each mass and the Tension in the string.  Sample Problem Determine the acceleration of each mass and the Tension in the string. +a Acceleration of the two masses is the same. m1 − m2 Assume positive acceleration a= g m1 + m2 is counterclockwise. 10 − 5 10 − 5 2 2 a= g= (9.8m/s ) = 3.27m/s 10 + 5 10 + 5  Sample Problem Determine the acceleration of each mass and the Tension in the string. Fnet = m1g − T Fnet = m1a T = m1g − m1a  Sample Problem Fnet = T − m2g Fnet = ma T = ma + m2g Checkpoint Checkpoint Checkpoint Checkpoint Checkpoint Checkpoint  Checkpoint Disclaimer: this equation is not the same as the one shown before and that is because we have ipped the direction of the acceleration. fl Checkpoint Checkpoint Checkpoint Checkpoint  Your Experiment In the lab you will measure the motion of both masses on the Atwood’s machine as the lighter one moves up and the heavier one moves down. You will use Capstone to calculate the changing speed of the masses as they move. It displays a graph of speed versus time. The acceleration of the system is the slope of speed versus time.  Motion on an Incline Plane Motion on an inclined plane refers to the movement of an object along a surface that is tilted at an angle θ relative to the horizontal. Inclined planes make work easier. They are used to move loads between high and low places.  Motion on an Incline Plane Let’s analyze the motion of a package on an inclined plane. What are the forces acting on it?  Motion on an Incline Plane Let’s analyze the motion of a package on an inclined plane. What are the forces acting on it? First you have gravity directly downwards.  Motion on an Incline Plane Let’s analyze the motion of a package on an inclined plane. What are the forces acting on it? First you have gravity directly downwards. You also have the normal force: the force the surface exerts on the object and is perpendicular to the plane's surface. If there is no friction, then this is all.  Motion on an Incline Plane Let’s analyze the motion of a package on an inclined plane. What are the forces acting on it? If there is no friction, then this is all. FN = Fg FN = mg cos θ θ Fnet = ma = mg sin θ θ  Motion on an Incline Plane If we add friction, the force of friction is given by: f = μN Here μ is the f = μmg cos θ coe cient of friction. The net acceleration then becomes: ffi Sample Problem Sample Problem Sample Problem Checkpoint Checkpoint Checkpoint Checkpoint  Your experiment You will use an inclined dynamics track will be used to determine the acceleration of a cart down the incline for different incline angles. For each angle, you will measure the velocity at 2 different points on the track and from there determine the acceleration. For this experiment, all the angles of incline are considered small enough such that cos θ ≈ 1. Thus a = g(sin θ − μ) 2 2 Also keep in mind the basic kinematic equation: v2 = v1 − 2aΔx You will use your measurements to determine the relationship between the acceleration and the incline angle.