(prev) | (top) | (next) |

- If we define the vector
**momentum****p**≡ m**v**,**force**experienced by an object (measured in**Newtons**: 1 N ≡ 1 kg m/s^{2}) is**F**≡ d**p**/dt= dm/dt

We will refer to this equation as "Newton's laws". In most cases, dm/dt = 0.**v**+ m**a** - Commonly useful forces are:
**F**= - m g**j**This is the force of

**gravity**near the Earth's surface.**Contact forces**are those forces experienced by two objects which "push" against each other. When a contact force is exerted between two objects, they both experience the same magnitude of force, but in opposite directions.F

_{N}is the**normal force**, perpendicular to the surface on which an object moves (or rests). If an object has a non-zero component of acceleration in the same direction as gravity, the normal force will be the*effective weight*of the object.- Ropes which do not stretch have a constant
**tension**along their entire length. This tension exerts a force in the direction opposite to every point of connection. We will assume that our ropes (and pulleys) are massless. Objects connected by such a rope (with possible fixed pulleys in between) experience the same*magnitude*of acceleration. When objects are connected via movable pulleys, the magnitudes of their accelerations are related by integer multiples. - F = μ F
_{N}This is the force due to

**friction**, acting in the opposite direction of motion. μ is the**coefficient of friction**(**static coefficient of friction**if the object is not moving,**kinetic coefficient of friction**if it is). **F**= ρ V g**j**This is the

**buoyant force**exerted by a fluid on a sinking object. ρ is the density of the fluid and V is the volume of the object, so the magnitude of the force is equal to the weight of the displaced fluid (**Archimedes' principle**).**F**= - b**v**This is the force of

**drag**experienced by an object slowly moving in a fluid; b is the**drag coefficient**.If the object has high velocity, the drag force is proportional to the square of the speed.

**F**= - k Δ**x**This is the force exerted by a spring. k is the

**spring constant**and Δ**x**is the displacement from equilibrium (where the spring is not stretched). This is known as**Hooke's law**.We will assume that our springs are massless. Note that a spring exerts the same magnitude of force on each end, but in opposite directions.

- And of course the
**centripetal force**F = m v^{2}/ r (pointing toward the center of the circular motion) is responsible for keeping the object in circular motion.

- Take an inventory of all the forces acting on each object, in all perpendicular directions (x and y, or parallel and normal if an incline is
involved). Gravity is almost always present, and there is a nonzero force in every direction in which there is a nonzero acceleration, so ask yourself
what the acceleration is in every direction.
For every direction in which any force acts and the component of acceleration is zero, there

Then write the vector component equations for Newton's laws for*must*be more than one force acting along that direction!*each*object or point of connection. Make sure that the relative signs of the forces and the accelerations make physical sense. State specifically the direction of motion associated with positive acceleration for each object under consideration.Consider the following contraption:

- Mass 1 (m
_{1}) is acted upon by the forces of- gravity (down),
- tension T
_{1}(up), - buoyancy (up)
- and drag (up if m
_{1}is sinking, down if m_{1}is rising).

- Mass 2 (m
_{2}) is acted upon by- the force of gravity (down),
- the normal force exerted by the surface (up),
- the contact force between masses 2 and 3 (denoted F
_{23}; left if m_{2}is moving to the right, 0 otherwise), - tensions T
_{1}(left) and - T
_{2}(right), and - the force of friction with the surface (right if m
_{2}is moving to the left, left if m_{2}is moving to the right).

- Mass 3 (m
_{3}) is acted upon by- the force of gravity (down),
- the normal force exerted by the surface (up),
- the contact force between masses 2 and 3 (right if m
_{2}is moving right, 0 otherwise), and - the force of friction with the surface (left if m
_{3}is moving to the right, 0 otherwise).

- Mass 4 (m
_{4}) is acted upon by- the force of gravity (down, producing a component normal to the incline and a component parallel to the incline
which contributes to m
_{4}sliding down the incline), - tension T
_{2}(up the incline), - the force of friction with the surface (down the incline if m
_{4}is moving up the incline, up the incline if m_{4}is moving down the incline), and - the force due to the spring k (up the incline if the spring is compressed beyond its equilibrium position, down the incline if the spring is stretched beyond its equilibrium position, and 0 if the spring is at its equilibrium position).

- the force of gravity (down, producing a component normal to the incline and a component parallel to the incline
which contributes to m

mass x y parallel to the incline normal to the incline m _{1}T _{1}- m_{1}g + ρ V g - b v = m_{1}am _{2}T _{2}- T_{1}- μ N_{2}- F_{23}= m_{2}aN _{2}- m_{2}g = 0m _{3}F _{23}- μ N_{3}= m_{3}aN _{3}- m_{3}g = 0m _{4}k (x _{0}- x_{4}) + m_{4}g sin θ - μ N_{4}- T_{2}= m_{4}aN _{4}- m_{4}g cos θ = 0Note that x

_{4}is measured parallel to the incline, and that we have assumed that the spring is extended (x_{4}is negative) so that the force due to the spring is pulling m_{4}down the incline.This system of equations is actually a complicated set of differential equations, and is an example of how much easier it is to write down equations in physics than it is to solve them. To make the problem more tractable, we will make the surfaces frictionless (by setting μ to 0), empty the fluid to remove bouyancy and drag (by setting ρ and b to 0), and remove both m

_{3}and the spring (by setting m_{3}and k to 0). The resulting system is a set of three linear equations in three variables (T_{1}, T_{2}and a), with the solution- T
_{1}= m_{1}g (m_{2}+ m_{4}(1 + sin θ)) / (m_{1}+ m_{2}+ m_{4}) - T
_{2}= m_{4}g (m_{1}+ (m_{1}+ m_{2}) sin θ) / (m_{1}+ m_{2}+ m_{4}) - a = (m
_{4}sin θ - m_{1}) g / (m_{1}+ m_{2}+ m_{4})

_{1}> m_{4}sin θ, the direction of motion is reversed. And if m_{1}= m_{4}sin θ, the system is in**equilibrium**: the net forces are zero.As long as b and k are zero, this system is still a set of linear equations which we can solve. But if ρ, μ or m

_{3}are nonzero, we must look carefully at the signs. For μ > 0, the solution is- T
_{1}= m_{1}g (m_{2}(1 - μ) + m_{4}(1 + sin θ - μ cos θ) ) / (m_{1}+ m_{2}+ m_{4}) - T
_{2}= m_{4}g (m_{1}+ μ m_{2}+ (m_{1}+ m_{2}) (sin θ - μ cos θ)) / (m_{1}+ m_{2}+ m_{4}) - a = (m
_{4}sin θ - (m_{1}+ μ (m_{2}+ m_{4}cos θ)) g / (m_{1}+ m_{2}+ m_{4})

*reverse*the motion! - Mass 1 (m
- Some forces are the result of measurement in accelerating frames of reference:
- The
**centrifugal force**is the apparent force you experience (in the opposite direction to the centripetal force) because you are in a rotating coordinate system. - The
**Coriolis force**is the apparent force due to the rotation of the Earth: the tangential speed is a decreasing function of latitude. - Every local gravitational force is equivalent to measurement in an accelerated reference frame; this is the
**Einstein equivalence principle**.

- The
- Be sure to distinguish when gravity is in the same plane as centripetal motion, and when it is perpendicular to the centripetal motion.
- In the former case, ask whether the circular motion acts to make the object seem lighter or heavier.
- In the latter case, there is often an important angle whose tangent is the ratio of the centripetal and gravitational forces.

Consider an object undergoing centripetal motion on a banked track of incline angle θ:

**N**= m g cos θ - m v^{2}sin θ / r**F**= m v_{parallel}^{2}cos θ / r - m g sin θ - μ**N****F**= 0, or_{parallel}tan θ = (v

In the limit θ = 0, the no-slip condition yields^{2}- μ g r) / (g r - μ v^{2}).μ = v

^{2}/ (g r).

(prev) | (top) | (next) |

©2010, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.