Kinematics - Plain Local Schools

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Transcript Kinematics - Plain Local Schools

Dynamics – Ramps and Inclines
Unit #3 Dynamics
Objectives and Learning Targets
 Calculate parallel and perpendicular components of
an object’s weight to solve ramp problems.
 Resolve a vector into perpendicular components: both
graphically and
 Use vector diagrams to analyze mechanical systems
(equilibrium and non-equilibrium).
Unit #3 Dynamics
Inclined Planes & Ramps
 Inclined Plane - A plane set at an angle to the
horizontal, especially a simple machine used to raise
or lower a load by rolling or sliding.
Unit #3 Dynamics
Understanding Inclined Planes
•The key to understanding these situations is creating an accurate free body diagram after
choosing convenient x- and y-axes. Problem-solving steps are consistent with those
developed forNewton's 2nd Law.
•Let's take the example of a box on a ramp inclined at an angle of Θ with respect to the
horizontal. We can draw a basic free body diagram for this situation, with the force of
gravity pulling the box straight down, the normal force perpendicular out of the ramp, and
friction opposing motion (in this case pointing up the ramp).
Unit #3 Dynamics
Understanding Inclined Planes
•Once the forces acting on the box have been identified, we must be clever about our
choice of x-axis and y-axis directions. Much like we did when analyzing free falling objects
and projectiles, if we set the positive x-axis in the direction of initial motion (or the
direction the object wants to move if it is not currently moving), the y-axis must lie
perpendicular to the ramp's surface (parallel to the normal force). Let's re-draw our free
body diagram, this time superimposing it on our new axes.
Unit #3 Dynamics
Resolving to Components
•Unfortunately, the force of gravity on the box, mg, doesn't lie along one of the
axes. Therefore, it must be broken up into components which do lie along the xand y-axes in order to simplify our mathematical analysis. To do this, we can use
geometry to break the weight down into a component parallel with the axis of
motion (mg║) and a component perpendicular to the x-axis (mg┴) using the
Unit #3 Dynamics
Resolving to Components
• Using these equations, we can re-draw the free body diagram, replacing mg
with its components. Now all the forces line up with the axes, making it
straightforward to write Newton's 2nd Law Equations (FNETx and FNETy) and
continue with our standard problem-solving strategy.
• In the example shown with our modified free body diagram, we could write our
Newton's 2nd Law Equations for both the x- and y-directions as follows:
From this point, our problem becomes an exercise in algebra. If
you need to tie to two equations together to eliminate a variable,
don't forget the equation for the force of friction:
Unit #3 Dynamics
Sample Problem #1
Unit #3 Dynamics
Sample Problem #2
Unit #3 Dynamics
Sample Problem #3
Question: Three forces act on a box on an inclined
plane as shown in the diagram below. [Vectors are
not drawn to scale.] If the box is at rest, the net
force acting on it is equal to
1. the weight
2. the normal force
3. friction
4. zero
Answer: (4) zero. If the box is at rest, the acceleration
must be zero, therefore the net force must be zero.
Unit #3 Dynamics