What is the difference between a single pendulum motion and a circular motion

A single pendulum is a centripetal force generated by gravity.

Its period t = 2pai (l g).

PAI is pi.

L is the length of the single pendulum rope and G is the acceleration due to gravity.

g = period of circular motion t = 1 f

f is the rotational speed, such as f=1200r min, t=60 1200s=1 20s=

The centripetal force is an effect force that is the driving force that sustains the object in a circular motion or a single pendulum motion, in a circular motion, it only changes the direction of motion of the object.

Understanding of Centripetal Force (1) The centripetal force is named from the effect of the force, because it produces acceleration directed towards the center of the circle, so it is called the centripetal force. It is not some type of force that has a definite nature. On the contrary, a force of any nature can act as a centripetal force.

In fact, it can be a force of a certain nature, or a component of a force, or it can be the combined force of several forces of different properties pointing towards the center of the circle along the radius. (2) Why doesn't the centripetal force pull the object towards the center of the circle The object in circular motion, the direction of velocity must change at all times, in order to change the direction of the object's velocity, a certain amount of force is required, assuming that the object is not subjected to force, then it will not fly out along the tangent direction under the action of inertia? When the object moves in a circular motion, the magnitude of the centripetal force is exactly equal to the required force, so it has no "spare force" to pull the object towards the center of the circle.

In fact, if the centripetal force is greater than the value required to do the circular motion, it will indeed pull the object towards the center of the circle.

The differences between a double pendulum and a single pendulum are as follows:First, the characteristics of Changfeng are different: compound pendulum:

In the process of oscillation, the compound pendulum is only subject to the reaction force of gravity and the rotating shaft, while the gravitational moment plays the role of restoring the torque. Single pendulum: At a very small amplitude (angle), the period of the single pendulum does simple harmonic motion and the square root of the pendulum length.

proportional to the acceleration of gravity.

is inversely proportional to the square root of , independent of the amplitude and the mass of the pendulum.

2. Composition of different compound pendulums: the intersection point o of the rotating shaft of the compound pendulum with the overrigid center of mass c and perpendicular to the plane of the rotating shaft is called the fulcrum or suspension point. In the process of swinging, the compound pendulum is only affected by the reaction force of gravity and the rotating shaft, while the gravitational moment of the trouser car plays the role of restoring the torque.

Single pendulum: made up of idealized pendulums and cycloids.

Composition The cycloid is provided by the thin line of uncountable and non-retractable thread; The pendulum ball is denser, and the radius of the ball is much smaller than the length of the cycloid.

What is the single pendulum principle.

One of the particle vibration systems is the simplest pendulum. An object that oscillates back and forth around a suspension point is called a pendulum, but its period is generally related to the distribution of the shape, size and density of the object. But if the size of the small size of the mass is suspended on one end of the fixed length of l and can not be stretched on the string, pull the mass away from the equilibrium position, so that the angle between the fine hunger and the plumb line of the suspension point is less than 5 °, the reciprocating vibration of the mass after letting go, can be regarded as the vibration of the particle, its period t is only related to l and the local gravitational acceleration g, that is, and the mass, shape and amplitude of the mass have nothing to do with the magnitude, its motion state can be expressed by a simple harmonic vibration formula, called a single pendulum or mathematical pendulum.

If the angle of the vibration is greater than 5°, the period of the vibration will increase with the increase of the amplitude and it will not be a single pendulum. If the size of the pendulum ball is quite large, the quality of the rope can not be ignored, it becomes a compound pendulum (physical pendulum), and the period is related to the size of the pendulum ball.

What does a physical, simple harmonic motion displacement diagram look like? What about single pendulum movement? Thanks to the displacement diagram of the simple harmonic motion, which is the displacement of the vibration (the displacement of the relative equilibrium position) as a function of time, which is a sine or cosine curve.

When the pendulum angle is very small (generally less than 5 degrees, now the textbook says that it is less than 10 degrees), its vibration in a plane is simple harmonic motion, and its displacement diagram is also a sine function imitation banquet number or cosine function curve.

The period of the single pendulum movement is related to the pendulum length: the longer the pendulum length, the longer the period; The shorter the pendulum length, the shorter the period. The period of motion of a single pendulum has nothing to do with the quality of the pendulum.

Under the condition that the amplitude of motion is very small, the period of motion of the single pendulum has no relationship with the amplitude of motion. The specific relationship is as follows:

where: t - the period of the movement of the single pendulum; l - pendulum length; g – acceleration due to gravity.

The completion of a full vibration of a single pendulum is a cycle, such as starting from the maximum displacement point of the single pendulum swing, then the pendulum ball returns to this point again, which is a cycle. If you start from any point in the middle, the pendulum will return to this point for the second time, which is a cycle.

Hang a particle point with an absolutely rigid and constant length and negligible mass, and make periodic motion in the plumb plane under the action of gravity, which becomes a single pendulum. When a pendulum vibrates at an oscillation angle of less than 10 degrees, it can be approximated as simple harmonic motion.

Hang a particle with an absolutely flexible line with the same length and negligible mass, and make periodic motion in the plumb plane under the action of gravity, which becomes a single pendulum. When a pendulum oscillates at an angle of less than 5° (now generally considered to be less than 10°), it can be approximated as simple harmonic motion. Periodic formula for single pendulum motion:

t=2π√(l/g).where l is the pendulum length, and g is the acceleration due to local gravity.

When the period of the single pendulum is t=2s, the length of the pendulum is about 1m by the formula, and the single pendulum in this case is called the second pendulum. The second pendulum is commonly found on pendulum clocks.

Note: In the current high school stage, the case of the pendulum angle less than 10° is generally studied (that is, the approximation is regarded as simple harmonic motion), and the high school textbook only involves the speculation of the formula in the experiment, and does not involve the derivation of the single pendulum period formula (because it needs to involve advanced mathematics).

First of all, by Newtonian mechanics, the motion of a single pendulum can be described as follows:

The gravitational moment experienced by a single pendulum is: mm

glsinx.

where m is the mass, g is the acceleration due to gravity, l is the pendulum length, and x is the pendulum angle.

We want to get the pendulum angle x as a function of time to describe the motion of a single pendulum. It is not difficult to get from the relationship between moment and angular acceleration, mj

where jml 2 is the moment of inertia of a single pendulum

x''(The 2nd derivative of the swing angle with respect to time) is the angular acceleration.

So it was simplified. x''l

gsinx.If we choose the proportional coefficient appropriately in the above equation, we can reduce the constant l to g, and then shift the term to obtain the simplified equation of motion.

x''sinx

Because the equation of motion (differential equation) of a single pendulum is.

x''sinx

The standard simple harmonic vibration (e.g. spring oscillator) is. x''x

We know that equation (1) is a nonlinear differential equation, while equation (2) is a linear differential equation. Therefore, strictly speaking, the motion of a single pendulum described in equation (1) above is not a simple harmonic motion.

However, in the X comparison, there is approximately sin

xx。(The radian system is taken here.) i.e. when x

There is sinxx at 0

o(1)。Thus, Eq. (1) becomes Eq. (2), and the nonlinear motion of a single pendulum is linearly approximated as a simple harmonic motion.

Then say why 5°. Due to sin

The approximation of xx only holds when the angle is relatively small (this can be seen from the approximation of the image of the sinusoidal function near the origin), so it is only reasonable to convert (1) to (2) at small angles.

In fact 5° radian, sin

5°, the difference between the two is only a few thousandths of a thousandth, which is very close. In low-precision experiments, this systematic error is negligible (because the accidental error in the experimental operation is larger). However, if it is changed to 25°, the error is as high as three percent, and it should no longer be regarded as simple harmonic vibration.

Due to the nature of the sinusoidal function, this approximation is that the smaller the angle, the more accurate, and the larger the angle, the less accurate. If the angle is very large (e.g. 17% at 60 degrees), it cannot be said to be a simple harmonic vibration.

Because the restoring force and the displacement are in opposite directions.

High School Physics.

A light rope has a large mass under it.

Smaller weights.

Movement with a swing angle of less than 10° in a vertical plane is called a single pendulum motion.

The uniform circular motion is decomposed into two simple harmonic vibrations of the same frequency, the same amplitude, and the phase difference is 2, and the direction of vibration is perpendicular to each other.

x(t)=rcos(ωt)，vx(t)=-rsin(ωt)，ax(t)=-2rcos(ωt)=-2 x(t)；

y(t)=rsin(ωt)，vy(t)=ωrcos(ωt)，ay(t)=-2rsin(ωt)=-2y(t)；

Compare with the simple harmonic vibration force f=ma=-kx, it can be seen.

2=k/m。

Circular trajectory: x 2 y 2=r , a=r 2

Two simple harmonic vibrations of the same frequency do not necessarily synthesize into a uniform circular motion. Trajectories may be ellipses, circles, straight lines. (mainly adjusts the phase difference and amplitude).

The main thing is to analyze according to this, and then you can do it.

There are cycles and frequencies.

The rate of uniform circular motion does not change, but the rate of a single pendulum changes.

The magnitude of acceleration does not change, the magnitude changes, and so on.

Single pendulum is a partial circular motion with variable speed, and the common denominator is that they are all affected by the centripetal force!