NEWTONIAN MECHANICS
PROBLEM SOLVING SKILLS
MOTION IN ONE DIMENSION
For problems involving uniformly accelerated motion:
1. Complete a data table with the information given. Use the proper sign for the quantity represented
by the symbol in the data table. Examples:
- If a car is slowing down, then the rate of acceleration is negative.
- If an object in free fall was initially thrown downward, then the downward direction is taken to be
positive and both the initial velocity and the rate of acceleration are positive.
- If the object in free fall was given an initial upward motion, then the upward direction is taken to
be positive. This means that the initial upward velocity is positive but the rate of acceleration is
negative.
2. Determine which formula or combination of formulas can be used to solve the problem.
For problems related to graphical analysis where distance is a function of time:
1. The instantaneous velocity at a particular moment in time equals the slope of the tangent line
drawn to the curve at the point in question. If the slope is positive, the object’s speed is positive. If
the slope is negative, the object’s speed is negative. If the slope is zero, the object is not moving.
For problems related to graphical analysis where velocity is a function of time:
1. The area under the curve represents the distance traveled.
2. If the area cannot be solved by a combination of rectangles and/or triangles, then determine the
area of one block. Count the number of blocks in the time interval being considered. Multiply the
total number of blocks by the distance represented by one block. The product is the total distance
traveled during the time interval.
3. The instantaneous acceleration at a particular moment of time is determined by finding the slope
of a tangent line. Draw a tangent line to the graph at the point in question, find the slope of the
tangent line. If the slope is positive, the object is accelerating. If the slope is negative, the object is
decelerating. If the slope is zero, the object is traveling at constant speed.
MOTION IN TWO DIMENSIONS
For problems involving vector addition or subtraction:
1. Use a protractor and ruler to accurately represent each vector involved in the problem. Use an
appropriate scale in representing the vector.
A) Graphical Method:
Two Vectors: Use the Parallelogram Method and measure both the magnitude and direction of the
resultant.
Three or more Vectors: Use the Polygon Method and measure both the magnitude and direction of
the resultant.
B) Trigonometric Component Method:
1)
break each vector into x and y components.
2)
use the sign convention and assign a positive sign or a negative sign to the magnitude.
3)
determine the sum of the x components and repeat for the sum of the y components.
4)
use the Pythagorean theorem and simple trigonometry to solve for the magnitude and
direction of the resultant.
For problems involving projectile motion:
1. Complete a data table with the information given. Use the proper sign for the quantity represented
by the symbol in the data table depending on whether the object was initially moving upward or
downward. Use the same sign convention as in free fall problems.
2. Use the trigonometric component method to determine the x and y components of the initial
velocity.
3. Determine which formula or combination of formulas can be used to solve the problem.
NEWTON'S LAWS OF MOTION
For problems related to Newton’s second law of motion:
1. Complete a data table with the information given.
2. Draw an accurate, labeled free-body-diagram locating each of the forces acting on the object or
system of objects.
3. If the object is on an incline, the weight is replaced with components acting parallel and
perpendicular to the incline.
4. If a frictional force is involved, the magnitude of the force is related to the coefficient of friction
and the normal force.
5. Determine the magnitude of the net force acting on the object.
6. Use Newton’s second law to write an equation for the motion of each object in the system. Solve
for the acceleration of each object in the system.
7. If the problem involves uniform acceleration, then the kinematics equations developed for
uniformly accelerated motion can be used to determine the velocity, displacement, time, etc.
UNIFORM CIRCULAR MOTION
For problems involving centripetal acceleration relate to Newton’s second law of motion.
1. Complete a data table with the information given.
2. Draw an accurate, labeled free-body- diagram locating each of the forces acting on the object in
uniform circular motion. Take note of the force(s) that is/are causing the object to travel in a curved
path.
3. Determine the magnitude and direction of the net force acting on the object.
4. Use Newton’s second law and the concept of centripetal acceleration to solve the problem.
NEWTON'S LAW OF GRAVITY
For problems involving Newton’s universal law of gravitation:
1. Complete a data table with the information given.
2. Use the universal law of gravitation and, if necessary, Newton’s second law and the concept of
centripetal acceleration to solve the problem.
For problems involving Kepler’s third law:
1. Complete a data table with the information given.
2. Use Kepler’s third law to solve the problem.
WORK, ENERGY AND POWER
For problems involving the work-energy theorem:
1. Complete a data table with the information given.
2. Draw an accurate, labeled diagram locating all of the forces, both conservative and nonconservative, acting on the object.
3. Determine the magnitude and direction of the net force acting on the object and then determine the
net work done on the object.
4. Apply the work energy theorem and solve the problem.
For problems involving the law of conservation of energy:
1. Complete a data table with the information given.
2. Draw an accurate, labeled diagram locating all of the forces, both conservative and nonconservative, acting on the object.
3. Apply the law of conservation of energy and solve the problem.
For problems involving power:
1. Complete a data table with the information given.
2. Draw an accurate, labeled diagram locating all of the forces, both conservative and nonconservative, acting on the object.
2. Apply the formulas for power and solve the problem.
LINEAR MOMENTUM
For problems involving impulse-change of momentum:
1. Complete a data table with the information given.
2. Draw an accurate, labeled diagram locating all of the forces acting on the system.
3. If a net external force acts on the object(s), then the momentum of the system will change.
Determine the magnitude and direction of the net force.
4. Apply the impulse-momentum equation taking note that force and velocity are vectors and that
direction of the vector plays an important part in the solution.
For problems involving no external force acting on the system:
1. Use the law of conservation of momentum to solve the problem. Take note that momentum is a
vector quantity and must be considered in the solution.
For problems involving graphical integration:
1. Determine the sum of the partial and complete blocks that lie under the curve.
2. Determine the impulse represented by one block and multiply the impulse represented by one
block by the total number of blocks.
3. Use the impulse-momentum equation and solve the problem.
For problems involving perfectly elastic and completely inelastic collisions:
1. Complete a data table with the information given.
2. Determine which type of collision is described in the problem.
3. If the collision is completely inelastic and the objects stick together, use the law of conservation of
momentum to solve the problem.
4. If the collision is perfectly elastic use, both conservation of momentum and conservation of
mechanical energy. Each law produces an algebraic equation with two unknowns. The final velocity
of each object can be determined by solving the equations.
STATIC EQUILIBRIUM AND TORQUE
For problems where the forces are concurrent and the object is in static equilibrium:
1. Draw an accurate, labeled diagram locating the forces acting on the object or system of objects.
2. Resolve each force vector into x and y components.
3. Apply the first condition of equilibrium (sum of forces = 0) and solve the problem.
For problems involving torque:
1. Draw a labeled diagram locating the axis of rotation of the object.
2. Determine the magnitude of the force and the lever arm distance.
3. Solve for the magnitude of the torque. If the direction of rotation is clockwise the torque is
negative, if it is counterclockwise the torque is positive.
For problems where the forces are non-concurrent and the object or system of objects is in static
equilibrium:
1. Draw an accurate, labeled diagram locating the forces acting on the object or system of objects.
2. Resolve each force vector into x and y components.
3. Write an equation(s) using the first condition of equilibrium.
4. Select a convenient point for the axis of rotation. A convenient point is located at a position where
an unknown force acts.
5. Determine the direction of the rotation (CW or CCW) produced by the force about the axis of
rotation.
6. Write an equation using the second condition of equilibrium (sum of torques = 0)
7. Using the two conditions of equilibrium to solve the problem.
SIMPLE HARMONIC MOTION
For problems involving a spring undergoing SHM:
1. Complete a data table with the information given.
2. Use Hooke’s law to determine the force constant of the spring.
3. To determine the period, use the formula that relates the period to the force constant and the mass
of the object.
4. Determine the total energy of the system and use the law of conservation of energy to determine
the velocity of the object at any point in the motion.
For problems involving a simple pendulum undergoing SHM:
1. Complete a data table with the information given.
2. Use the formula for the pendulum period to solve for the period, length, or the magnitude of the
gravitational acceleration.
NEWTONIAN MECHANICS FREE RESPONSE AP PROBLEMS
The list of problems is given as follows:
Year of examination
Problem number
Topics covered in problem
1983, 1, torque and static rotation
1983, 2, SHM and momentum
1984, 1, centripetal force, kinematics in two dimensions
1984, 2, momentum
1985, 1, momentum, potential and kinetic energy
1985, 2, inclined plane, forces, work, potential and kinetic energy
1986, 1, Newton's second law
1986, 2, kinematics in two dimensions, elastic potential energy, work-energy theorem
1987, 1, Newton's second law, friction
1989, 1, centripetal force, kinematics in two dimensions
1990, 1, momentum, kinetic energy, kinematics in two dimensions
1991, 1, static equilibrium, conservation of energy, centripetal force
1992, 1, centripetal force, kinematics in two dimensions
1992, 2, conservation of momentum, conservation of energy
1994, 1, projectile motion, impulse-momentum
1994, 2, conservation of momentum, conservation of energy
1995, 1, conservation of momentum, conservation of energy, SHM
1995, 3, forces, Newton's second law, centripetal force
1996, 1, conservation of momentum, conservation of energy
1996, 2, experiment to determine spring constant
1997, 1, kinematics in one dimension, Newton's second law, work-energy theorem, impulsemomentum
1997, 2 uniform circular motion: horizontal circle
1997, 3 part a) Hooke's law, part d) conservation of elastic potential energy and kinetic energy
2000, 1, kinematics in one dimension, graphical analysis of motion
2000,2, inclined plane, Newton's second law
2001, 1, uniform circular motion: vertical circle
2001, 2, conservation of momentum, conservation of energy, kinematics in two dimensions
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