we found that a particle moving with uniform speed v in a circular path
of radius r experiences an acceleration that has a magnitude
The acceleration is called centripetal acceleration because is directed toward the center of the circle. Furthemore is always perpendicular to v. ( if there were a component of acceleration paralel to v, the particle’s speed would be changing)
Consider a ball of mass m that is tied to a string of length r and is being whirled at constant speed in a horizonal circular path, as illustrated in figure 6.1 its weight is supported by a frictionless table. Why does the ball move in circle? According to Newton’s firs law, the ball tends to move in a straight line. However, the string prevents motion along a straight line by exening on the ball radial force that makes it follow the circular path. This force is directed along the string toward the center of the circle as shown in figure 6.1
If we apply Newton’s law along the radial direction, we find that the net force causing the centripetal acceleration can be evaluated :
A force causing a centripetal acceleration acts toward the center of the circular path and causes a change in the direction of the velocity vector.If that force should vanish, the object would no longer move in its circular path: instead. It would move a long a straight-line path tangent to the circle. This idea is illustrated in figure 6.2 for the ball whirling at the end of a sting in horizontal plane. If the string breaks at some instant, the ball moves along the straight-line path tangen to the circle at the point where the string breaks
NONUNIFORM CIRCULAR MOTTION
We found that if a particle moves with varying speed in a circular path,there is in addition to the radial component having magnitude Therefore,the force acting on the particle must also have a tangential and a radial component. Because the total acceleration is
( at, the total force exerted on the particle is , as shown in, Figure 6.8. The vector is directed toward the center of the circle and is responsible for the centripetal acceleration. The vector tangent to the circle is responsible for the tangential acceleration, which represents a change in the speed of the particle with time.
MOTION OF A SYSTEM OF PARTICLES
Ò a force has acted on the puck to cause it to accelerate called a fictitious force. Fictitious forces may not be real forces, but they can have real effects.
Ò a fictitious is due to an accelerated reference frame. While the real forces are always due to interactions between two objects
Ò Real forces are always due to interactions between two objects. A fictitious force appears to act on an object in the same way as a real force, but you can’t identify a second object for a fictitious force.
A fictitious force is due to the change in the direction of the velocity vector. To understand the motion of a system that is noninertial because of a change in direction, consider a car traveling along a highway at a high speed and approaching a rection, curved exit ramp, as shown in Figure 6.11a. As the car takes the sharp left turn onto the ved ramp, a person sitting in the passenger seat slides to the right and hits the door door. At that point, the force exerted by the door on the passenger keeps her from being ejected from the car car. What causes her to move toward the door? A popular but incorrect explanation is that a force acting toward the right in Figure 6.11b pushes her outward. This is often called the “centrifugal force” but it is a fi fictitious force due to the acceleration associated ctitious with the changing direction of the car car’s velocity vector vector. (The driver also experiences this effect but wisely holds on to the steering wheel to keep from sliding to the right.)
Another fictitious force is the “Coriolis force” This is an apparent force caused by changing the radial position of an object in a rotating coordinate system. For example, suppose you and a friend are on opposite sides of a rotating circular platform and you decide to throw a baseball to your friend. As Figure 6.12a shows, at t = 0 you throw the ball toward your friend, but by the time tf when the ball has crossed the plat- platform, your friend has moved to a new position.
Active Figure 6.12 (a) You and your friend sit at the edge of a rotating turntable. In this overhead view observed by someone in an inertial reference frame attached to the ved Earth, you throw the ball at t ! 0 in the direction of your friend. By the time tf that the ball arrives at the other side of the turntable, your friend is no longer there to catch it. According to this obser observer ver, the ball followed a straight line path, consistent with, Newton Newton’s laws. (b) From the point of view of your friend, the ball veers to one side during its fl flight. Y ight. Your friend introduces a our fictitious force to cause this deviation from fictitious the expected path. This fictitious force is called the “Coriolis force.”
Fictitious forces may not be real forces, but they can have real effects. An object on your dashboard really slides off if you press the accelerator of your car. As you ride on a merry-go-round, you feel pushed toward the outside as if due to the fictitious “centrifugal force.” You are likely to fall over and injure yourself if you walk along a radial line while merry-go-round rotates. The Coriolis force due to the rotation of the Earth is responsible for rotations of hurricanes and for large-scale ocean currents.
MOTION THE PRESENCE OF RESISTIVE FORCE
The magnitude of the resistive force can depend on speed in a complex way, and here we consider only two situations.
In the first situation, we assume the resistive force is proportional to the speed of the moving object; this assumption is valid for objects falling slowly through a liquid and for very small objects, such as dust particles, moving through air.
In the second situation, we assume a resistive force that is proportional to the square of the speed of the moving object; large objects, such as a skydiver moving through air in free fall, experience such a force.
1. Resistive Force Proportional to Object Speed
If we assume that the resistive force acting on an object moving through a liquid or gas is proportional to the object object’s speed, then the resistive force can be expressed as:
where v is the velocity of the object and b is a constant whose value depends on the
properties of the medium and on the shape and dimensions of the object. If the object
is a sphere of radius r, then , b is proportional to r. The negative sign indicates that . R is in the opposite direction to v. Consider a small sphere of mass m released from rest in a liquid, as in Figure 6.15a. Assuming that the only forces acting on the sphere are the resistive force
2. Air Drag at High Speed
For objects moving at high speeds through air air, such as airplanes, sky divers, cars, and baseballs, the resistive force is approximately proportional to the square of the speed. In these situations, the magnitude of the resistive force can be expressed as:
ρ = the density of air,
A = the cross-sectional area
D = the drag coefficient
spherical objects= 0.5
irregularly shaped objects=2
V = the velocity
R = the resistive force
NUMERICAL MODELING DYNAMIC
If a particle of mass m moves under the in infl fluence of a net force Newton’s second law tells us that the acceleration of the particle is In general, we apply te analytical method to a dynamics problem using the following procedure:
1. Sum all the forces acting on the particle to fi find the net force
2. Use this net force to determine the acceleration from the relationship.
3. Use this acceleration to determine the velocity from the relationship dv/dt= a
4. Use this velocity to determine the position from the relationship dx/dt= v
The Euler Method
One advantage of the Euler method is that the dynamics is not obscured —the fundamental relationships between acceleration and force, velocity and acceleration,
and position and velocity are clearly evident. Indeed, these relationships form the
heart of the calculations. There is no need to use advanced mathematics, and the basic
physics governs the dynamics.
The Euler method is completely reliable for infinitesimally small time increments, but for practical reasons a finite increment size must be chosen. For the finite difference
approximation of Equation 6.10 to be valid, the time increment must be small enough that the acceleration can be approximated as being constant during the incre- increment. We can determine an appropriate size for the time increment by examining theparticular problem being investigated. The criterion for the size of the time incrementmay need to be changed during the course of the motion. In practice, however however, we usu- , usuallychoose a time increment appropriate to the initial conditions and use the same value throughout the calculations.
The size of the time increment in infl fluences the accuracy of the result, but unfortuences unfortunately. it is not easy to determine the accuracy of an Euler nately Euler-method solution without a knowledge of the correct analytical solution. One method of determining the accuracy of the numerical solution is to repeat the calculations with a smaller time increment and compare results. If the two calculations agree to a certain number of signicant figures, you can assume that the results are correct to that precision.