$3.5.3^*.$ Under the conditions of problem 3.5.1, the pendulum had a velocity $v_0$ and a coordinate $x_0$ at the initial moment ($t=0$). What will be the amplitude of oscillations after $n$ impacts if the first one occurred at the initial moment? Plot the phase portrait.
*(Note: in the answer section of the problem book, the transferred momentum is denoted as $p$, and not $p_0$ as in the condition of problem 3.5.1. For consistency with the author's answer, the notation $p$ is used below).*
### Solution
From problem 3.5.2, it is known that simple harmonic motion on the phase plane with coordinates $(x, \frac{p_{\text{imp}}}{m\omega})$ corresponds to motion along a circle. The radius of this circle is numerically equal to the oscillation amplitude $A$.
By the Pythagorean theorem, the amplitude for any state is expressed as:
$$ A = \sqrt{x^2 + \left(\frac{p_{\text{imp}}}{m\omega}\right)^2}$$
It is important to note that at the moment of a brief impact, the coordinate $x$ of the pendulum does not have time to change, but its momentum increases abruptly by the value of the transferred momentum $p$.
Let us consider two cases for the impact frequency.
#### Case 1: Impacts follow each other after a period $T_0$
Over a time equal to the period of free oscillations $T_0 = 2\pi/\omega$, the representative point on the phase portrait makes a full revolution (a rotation by $360^\circ$) and returns to the same coordinates.
1. At the initial moment ($t=0$), the initial momentum is $mv_0$. Immediately after the first impact, the momentum becomes:
$$ p_1 = mv_0 + p $$
2. By the time of the second impact (after a time $T_0$), the pendulum completes a full oscillation and returns to the point with coordinate $x_0$, and its momentum is again equal to $p_1$. The second impact instantly adds another momentum $p$:
$$ p_2 = (mv_0 + p) + p = mv_0 + 2p $$
3. Since every period the system returns to its initial position $x_0$, the momenta simply accumulate algebraically. Immediately after the $n$-th impact, the total momentum at this point will be:
$$ p_n = mv_0 + np $$
4. Substituting the coordinate $x_0$ and the accumulated momentum $p_n$ into the amplitude formula, we obtain:
@@ -35,7 +28,7 @@Case 1: Impacts follow each other after a period $T_0$
On the phase portrait, this motion looks like an unwinding spiral consisting of concentric circular arcs of increasing radius.
−
+
#### Case 2: Impacts follow each other after half a period $T_0/2$
Over the time $T_0/2$, the point on the phase plane makes half a revolution (a rotation by $180^\circ$). In this process, both coordinates (both position and momentum) change their sign to the opposite.
1. First impact ($n=1$, odd): Occurs at $t=0$. The momentum abruptly becomes $p_1 = mv_0 + p$. The amplitude after the first impact is:
2. Free oscillations until the second impact: After the time $T_0/2$, both coordinates change sign. The position becomes $-x_0$, and the momentum becomes $-(mv_0 + p)$.
3. Second impact ($n=2$, even): An impact is delivered, adding a momentum $+p$. We calculate the new momentum:
$$ p_2 = -(mv_0 + p) + p = -mv_0 $$
Let us calculate the amplitude for this state (taking into account the coordinate $-x_0$):
The pendulum "sheds" the energy of the impact and transitions to an orbit of a smaller radius (the amplitude decreases).
4. Free oscillations until the third impact: After another $T_0/2$ (a full period $T_0$ has passed in total), the coordinates change sign again: $-(-x_0) = x_0$ and $-(-mv_0) = mv_0$. The system has returned exactly to its initial state (which was before the first impact).
5. Third impact ($n=3$, odd): Once again adds a momentum $p$ to the initial $mv_0$. The state becomes completely identical to the state after the first impact, therefore, $A_3 = A_1$.
@@ -54,7 +47,7 @@Case 2: Impacts follow each other after half a period $T_0/2$
Thus, for all odd $n$, the amplitude is equal to $A_1$, and for all even $n$, it is equal to $A_2$. On the phase portrait, the representative point infinitely jumps between two closed semicircles (outer and inner).
−
+
### Answer
If the impacts follow each other at time intervals $T_0$, then the amplitude is:
@@ -1,11 +1,4 @@
### Statement
### Statement
$3.5.3.$ [Insert the problem statement]
### Solution
### Problem
$3.5.3^*.$ Under the conditions of problem 3.5.1, the pendulum had a velocity $v_0$ and a coordinate $x_0$ at the initial moment ($t=0$). What will be the amplitude of oscillations after $n$ impacts if the first one occurred at the initial moment? Plot the phase portrait.
$3.5.3^*.$ Under the conditions of problem 3.5.1, the pendulum had a velocity $v_0$ and a coordinate $x_0$ at the initial moment ($t=0$). What will be the amplitude of oscillations after $n$ impacts if the first one occurred at the initial moment? Plot the phase portrait.
*(Note: in the answer section of the problem book, the transferred momentum is denoted as $p$, and not $p_0$ as in the condition of problem 3.5.1. For consistency with the author's answer, the notation $p$ is used below).*
*(Note: in the answer section of the problem book, the transferred momentum is denoted as $p$, and not $p_0$ as in the condition of problem 3.5.1. For consistency with the author's answer, the notation $p$ is used below).*
### Solution
### Solution
From problem 3.5.2, it is known that simple harmonic motion on the phase plane with coordinates $(x, \frac{p_{\text{imp}}}{m\omega})$ corresponds to motion along a circle. The radius of this circle is numerically equal to the oscillation amplitude $A$.
From problem 3.5.2, it is known that simple harmonic motion on the phase plane with coordinates $(x, \frac{p_{\text{imp}}}{m\omega})$ corresponds to motion along a circle. The radius of this circle is numerically equal to the oscillation amplitude $A$.
By the Pythagorean theorem, the amplitude for any state is expressed as:
By the Pythagorean theorem, the amplitude for any state is expressed as:
$$ A = \sqrt{x^2 + \left(\frac{p_{\text{imp}}}{m\omega}\right)^2}$$
$$ A = \sqrt{x^2 + \left(\frac{p_{\text{imp}}}{m\omega}\right)^2}$$
It is important to note that at the moment of a brief impact, the coordinate $x$ of the pendulum does not have time to change, but its momentum increases abruptly by the value of the transferred momentum $p$.
It is important to note that at the moment of a brief impact, the coordinate $x$ of the pendulum does not have time to change, but its momentum increases abruptly by the value of the transferred momentum $p$.
Let us consider two cases for the impact frequency.
Let us consider two cases for the impact frequency.
#### Case 1: Impacts follow each other after a period $T_0$
#### Case 1: Impacts follow each other after a period $T_0$
Over a time equal to the period of free oscillations $T_0 = 2\pi/\omega$, the representative point on the phase portrait makes a full revolution (a rotation by $360^\circ$) and returns to the same coordinates.
Over a time equal to the period of free oscillations $T_0 = 2\pi/\omega$, the representative point on the phase portrait makes a full revolution (a rotation by $360^\circ$) and returns to the same coordinates.
1. At the initial moment ($t=0$), the initial momentum is $mv_0$. Immediately after the first impact, the momentum becomes:
1. At the initial moment ($t=0$), the initial momentum is $mv_0$. Immediately after the first impact, the momentum becomes:
$$ p_1 = mv_0 + p $$
$$ p_1 = mv_0 + p $$
2. By the time of the second impact (after a time $T_0$), the pendulum completes a full oscillation and returns to the point with coordinate $x_0$, and its momentum is again equal to $p_1$. The second impact instantly adds another momentum $p$:
2. By the time of the second impact (after a time $T_0$), the pendulum completes a full oscillation and returns to the point with coordinate $x_0$, and its momentum is again equal to $p_1$. The second impact instantly adds another momentum $p$:
$$ p_2 = (mv_0 + p) + p = mv_0 + 2p $$
$$ p_2 = (mv_0 + p) + p = mv_0 + 2p $$
3. Since every period the system returns to its initial position $x_0$, the momenta simply accumulate algebraically. Immediately after the $n$-th impact, the total momentum at this point will be:
3. Since every period the system returns to its initial position $x_0$, the momenta simply accumulate algebraically. Immediately after the $n$-th impact, the total momentum at this point will be:
$$ p_n = mv_0 + np $$
$$ p_n = mv_0 + np $$
4. Substituting the coordinate $x_0$ and the accumulated momentum $p_n$ into the amplitude formula, we obtain:
4. Substituting the coordinate $x_0$ and the accumulated momentum $p_n$ into the amplitude formula, we obtain:
@@ -35,7 +28,7 @@Case 1: Impacts follow each other after a period $T_0$
On the phase portrait, this motion looks like an unwinding spiral consisting of concentric circular arcs of increasing radius.
On the phase portrait, this motion looks like an unwinding spiral consisting of concentric circular arcs of increasing radius.
#### Case 2: Impacts follow each other after half a period $T_0/2$
#### Case 2: Impacts follow each other after half a period $T_0/2$
Over the time $T_0/2$, the point on the phase plane makes half a revolution (a rotation by $180^\circ$). In this process, both coordinates (both position and momentum) change their sign to the opposite.
Over the time $T_0/2$, the point on the phase plane makes half a revolution (a rotation by $180^\circ$). In this process, both coordinates (both position and momentum) change their sign to the opposite.
1. First impact ($n=1$, odd): Occurs at $t=0$. The momentum abruptly becomes $p_1 = mv_0 + p$. The amplitude after the first impact is:
1. First impact ($n=1$, odd): Occurs at $t=0$. The momentum abruptly becomes $p_1 = mv_0 + p$. The amplitude after the first impact is:
2. Free oscillations until the second impact: After the time $T_0/2$, both coordinates change sign. The position becomes $-x_0$, and the momentum becomes $-(mv_0 + p)$.
2. Free oscillations until the second impact: After the time $T_0/2$, both coordinates change sign. The position becomes $-x_0$, and the momentum becomes $-(mv_0 + p)$.
3. Second impact ($n=2$, even): An impact is delivered, adding a momentum $+p$. We calculate the new momentum:
3. Second impact ($n=2$, even): An impact is delivered, adding a momentum $+p$. We calculate the new momentum:
$$ p_2 = -(mv_0 + p) + p = -mv_0 $$
$$ p_2 = -(mv_0 + p) + p = -mv_0 $$
Let us calculate the amplitude for this state (taking into account the coordinate $-x_0$):
Let us calculate the amplitude for this state (taking into account the coordinate $-x_0$):
The pendulum "sheds" the energy of the impact and transitions to an orbit of a smaller radius (the amplitude decreases).
The pendulum "sheds" the energy of the impact and transitions to an orbit of a smaller radius (the amplitude decreases).
4. Free oscillations until the third impact: After another $T_0/2$ (a full period $T_0$ has passed in total), the coordinates change sign again: $-(-x_0) = x_0$ and $-(-mv_0) = mv_0$. The system has returned exactly to its initial state (which was before the first impact).
4. Free oscillations until the third impact: After another $T_0/2$ (a full period $T_0$ has passed in total), the coordinates change sign again: $-(-x_0) = x_0$ and $-(-mv_0) = mv_0$. The system has returned exactly to its initial state (which was before the first impact).
5. Third impact ($n=3$, odd): Once again adds a momentum $p$ to the initial $mv_0$. The state becomes completely identical to the state after the first impact, therefore, $A_3 = A_1$.
5. Third impact ($n=3$, odd): Once again adds a momentum $p$ to the initial $mv_0$. The state becomes completely identical to the state after the first impact, therefore, $A_3 = A_1$.
@@ -54,7 +47,7 @@Case 2: Impacts follow each other after half a period $T_0/2$
Thus, for all odd $n$, the amplitude is equal to $A_1$, and for all even $n$, it is equal to $A_2$. On the phase portrait, the representative point infinitely jumps between two closed semicircles (outer and inner).
Thus, for all odd $n$, the amplitude is equal to $A_1$, and for all even $n$, it is equal to $A_2$. On the phase portrait, the representative point infinitely jumps between two closed semicircles (outer and inner).
### Answer
### Answer
If the impacts follow each other at time intervals $T_0$, then the amplitude is:
If the impacts follow each other at time intervals $T_0$, then the amplitude is:
