Angular Motion 彎道超車

Conservation of momentum Momentum p: Example: An object of mass M flies with velocity v splits into two pieces of equal mass M/2 because of the occurrence of an internal explosion. If one piece has velocity u, what is the velocity of the other piece right after the explosion?

Angular velocity l Radian(弧度): R Angular speed:

Angular momentum where I is the moment of inertia.

Moment of inertia A thin rod of length l Through the center A thin rod of length l Through one end Sphere of radius r Along a diameter Cylinder of radius r Along axis of symmetry

Conservation of angular momentum

Centripetal force （向心力） Figure 6.1  Overhead view of a ball moving in a circular path in a horizontal plane. A force Fr directed toward the center of the circle keeps the ball moving in its circular path.

Centrifugal force? 離心力? Active Figure 6.2  Overhead view of a ball moving in a circular path in a horizontal plane. When the string breaks, the ball moves in the direction tangent to the circle. At the Active Figures link at http://www.pse6.com, you can “break” the string yourself and observe the effect on the ball’s motion.

Force on a Curved Path The simplest angular motion is one in which the body moves along a curved path at a constant velocity, as when a runner travels along a circular path or an automobile rounds a curve. The usual problem here is to calculate the centripetal forces and determine their effect on the motion of the object. A common problem to raise is the maximum speed at which an automobile can round a curve without skidding.

The centripetal force Fc exerted on the moving car is Consider a car of weight W moving on a curved level road that has a radius of curvature R. The centripetal force Fc exerted on the moving car is For the car to remain on the curved path, a centripetal force must be provided by the frictional force between the road and the tires. The car begins to skid on the curve when the centripetal force is greater that the frictional force. Figure 6.5  (a) The force of static friction directed toward the center of the curve keeps the car moving in a circular path. (b) The free-body diagram for the car.

Force on a Curved Path Safe speed on a curved path may be increased by banking the road along the curve. In the absence of friction, the reaction force Fn acting on the car must be perpendicular to the road surface. The vertical component of this force supports the weight of the car. That is, To prevent skidding on a frictionless surface, the total centripetal force must be provided by the horizontal component of Fn;

v = ? Stable angle of a conic pendulum Figure 6.4   The conical pendulum and its free-body diagram.

Stable angle of a conic pendulum Figure 6.4   The conical pendulum and its free-body diagram.

A Runner on a Curved Track As the runner rounds the curve, he leans toward the center of rotation.

A Runner on a Curved Track There are two forces acting on the runner. W The upward force. Fcp The centripetal reaction force. The proper angle for a speed of 6.7 m/sec on a 15-m radius is

F = ? 如果飛機保持等速率, 座椅作用在駕駛員的力量,何處最大? 如果座椅就是磅秤, 駕駛員在何處最重? Figure 6.7  (a) An aircraft executes a loop-the-loop maneuver as it moves in a vertical circle at constant speed. (b) Free-body diagram for the pilot at the bottom of the loop. In this position the pilot experiences an apparent weight greater than his true weight. (c) Free-body diagram for the pilot at the top of the loop.

Pendulum Since the limbs of animals are pivoted at the joint, the swinging motion of animals is basically angular. Many of the limb movements in walking and running can be analyzed in terms of the swinging movement of a pendulum. WALKING RUNNING

Pendulum Tiger Woods golf swing

Pendulum A simple pendulum consists of a weight attached to a string, the other end of which is attached to a fixed point. If the pendulum is displaced a distance A from the center position and then released. It will swing back and forth under force of gravity. Such kind of back and forth movement is called a simple harmonic motion. g Equation of motion: If θ is very small then we have The solution is where is the angular frequency of the system. Period

At the extreme of the swing, the pendulum is momentarily stationary. As the pendulum swings, there is continuous interchange between potential and kinetic energies. At the extreme of the swing, the pendulum is momentarily stationary. Here its energy is entirely in the form of potential energy. g FT At the extreme of the swing where amax is the maximum tangential acceleration. Since and

Pendulum As the pendulum is accelerated to the center, its velocity increases, and the potential energy is converted to kinetic energy. The pendulum reaches its maximum velocity vmax. g FT Potential energy at the maximum angle = Kinetic energy at the lowest position At small angle θ

Walking Some aspects of walking can be analyzed in terms of the simple harmonic motion of a pendulum. The motion of one foot in each step can be considered as approximately a half-cycle of a simple harmonic motion. A person walks at a rate of 2 steps/sec and that each step is 90 cm long. In the process of walking each foot rests on the ground for 0.5 sec (T = 1 sec) and then swings forward 180cm and comes to rest again 90 cm ahead of the other foot. The speed of walking v is The maximum velocity of the swinging foot vmax is This is three times faster than the body.

Physical Pendulum In the discussion of a simple pendulum the total mass of the system is assumed to be located at the end of the pendulum. A more realistic model of a pendulum should be a physical pendulum, which takes into account that the distribution of mass along the swinging object. O It can be shown that under the force of gravity the period of oscillation T for a physical pendulum is Here I is the moment of inertia of the pendulum around the pivot point O. This quantity should be calculated by some integration. The distance of the center of gravity from the pivot point O is described as “r”.

Speed for Walking Walking and running are two different types of human motions. Walking In the analysis of walking and running, the leg may be regarded as a physical pendulum. The moment of inertia I for the leg is where W is the weight of the leg and l is its length If we assume that the center of mass of the leg is at its middle (r = 0.5 l), the period of oscillation is The period of walking is ~ l1/2 For a 90-cm-long leg, the period is 1.6 second. The most effortless walking speed for this person if the length of each step is 90cm is:

Speed for Walking Each step in the act of walking can be regarded as a half-swing of a simple harmonic motion, the number of steps per second is simply the inverse of the half period. In a most effortless walk, the legs swing at their natural frequency, and the time for one step is T/2. The speed of walking ∝ Number of steps × The length of the step The speed of walking v l: the length of the leg Since Thus, the speed of the natural walk of a person increases as the square root of the length of his legs.

Speed for Running The situation is different when a person (or any animal) runs at full speed. Whereas in a natural walk the swing torque is produced primarily by gravity, in a fast run the torque is produced mostly by the muscles. Using some reasonable assumptions, we can show that similarly built animals can run at the same maximum speed, regardless of differences in leg size.

The mass of the leg is proportional to l3. Speed for Running We assume that the length of the leg muscles is proportional to the length of the leg (l) and that the area of the leg muscles is proportional to l2. The mass of the leg is proportional to l3. The maximum force that a muscle can produce (Fm) is proportional to the area of the muscle. The maximum torque produced buy the muscle is proportional to the product of the force and the length of the leg; that is,

I: the moment of inertia Speed for Running In general, the period of oscillation for a physical pendulum under the action of a torque with maximum value of is given by I: the moment of inertia Since The maximum speed of running vmax is again proportional to the product of the number of steps per second and the length of the step. Because the length of the step is proportional to the length of the leg, we have

A fox can run as fast as a horse. This shows that the maximum speed of running is independent of the leg size, which is in accordance with observation. A fox can run as fast as a horse.

Exercise: If a person stands on a rotation pedestal with his arms loose, the arms will rise toward a horizontal position. (a) Explain the reason for this phenomenon. (b) Calculate the angular (rotational) velocity of the pedestal for the angle of the arm to be at 60 with respect to the horizontal. What is the corresponding number of revolutions per minute? Assume that the length of the arm is 90 cm and the center of mass is at mid-length. Answer: w = 3.546 rad/s = 33.9 rpm