BackForces, Free-Body Diagrams, and Newtonian Mechanics: Study Notes
Study Guide - Smart Notes
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Forces and Acceleration in Newtonian Mechanics
Vector Addition of Forces
When multiple forces act on an object, their vector sum determines the net force, which in turn governs the object's acceleration according to Newton's Second Law.
Key Point 1: Force vectors can be expressed in unit vector notation, such as .
Key Point 2: Net force is the sum of all individual forces acting on an object.
Formula: Newton's Second Law:
Example: If two forces and act on a block, the net force is .
Free-Body Diagrams
A free-body diagram is a graphical representation showing all the forces acting on an object. It is essential for analyzing the dynamics of the system.
Key Point 1: Each force is represented as an arrow pointing in the direction of the force, with length proportional to its magnitude.
Key Point 2: Common forces include gravity, normal force, friction, tension, and applied forces.
Example: For a block on a horizontal surface, the free-body diagram includes the normal force upward, gravity downward, and any applied horizontal forces.
Motion on a Horizontal Surface
Horizontal Forces and Acceleration
When an object moves on a horizontal surface with negligible friction, the net horizontal force determines its acceleration.
Key Point 1: If two horizontal forces act at angles, their vector sum must be calculated to find the net force.
Key Point 2: The acceleration is found using , where is the object's mass.
Example: If along the x-axis and at north of west, resolve into components before summing.
Variable Forces and Kinematics
Motion Under a Variable Force
When a force varies with time or position, the acceleration and velocity of the object must be determined using calculus or kinematic equations.
Key Point 1: The position function can be given as a polynomial in ; its derivatives yield velocity and acceleration.
Formula: ,
Example: For , ,
Systems of Masses and Tension
Multiple Masses Connected by Cords
When masses are connected by cords and pulleys, the system's acceleration and the tension in the cords can be analyzed using Newton's laws.
Key Point 1: Atwood's machine consists of two masses connected over a pulley; the acceleration depends on the difference in their weights.
Formula: For masses and ,
Key Point 2: The tension in the cord is
Example: If and , substitute into the formulas above.
Three Masses and Pulleys
For systems with three masses and multiple cords, analyze each mass separately and apply Newton's laws to each.
Key Point 1: Draw free-body diagrams for each mass, showing all forces.
Key Point 2: Write equations for the acceleration and tension in each cord.
Formula: For block A, (Additional info: This formula is inferred for a typical three-mass pulley system.)
Example: Given , , , calculate the acceleration and tension.
Free-Body Diagrams for Rotational Systems
Disks Suspended by Cords
When disks are suspended by cords, analyze the forces acting on each disk to determine the tension and mass relationships.
Key Point 1: Each disk experiences gravitational force downward and tension upward.
Key Point 2: The tension in the cord equals the weight of the disks below it.
Example: If , , , the mass of each disk can be found by .
Table: Tension and Mass Relationships in Suspended Disks
Disk | Tension (N) | Mass (kg) |
|---|---|---|
D | 60 | 6.12 |
C | 50 | 5.10 |
B | 10 | 1.02 |
A | 0 | 0 |
Additional info: Masses calculated using , assuming .
