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<meta name="description" content="A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?">
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<meta property="og:title" content="A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?">
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<meta property="og:description" content="A plane capacitor is placed in a homogeneous magnetic field od induction B parallel to the plates. From point A, electrons enter perpendicularly to magnetic field direction. The tension (voltage) applied to the plates is V. What is the condition for which electrons pass through the capacitor?">
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<meta property="og:description" content="A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?">
<title>A plane capacitor is placed in a homogeneous magnetic field od induction B parallel to the plates. From point A, electrons enter perpendicularly to magnetic field direction. The tension (voltage) applied to the plates is V. What is the condition for which electrons pass through the capacitor?</title>
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<title>A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?</title>
A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?
</p>
<h3>Solution</h3>
<p>
For a harmonic motion, variable position depends sinusoidally on time. In this case, we suppose that load is at equilibrium position at $t = 0$. So, sine function is more adequate to this situation without phase angle or null phase angle. So,
$$x(t) = A~\sin{~\omega t}$$
where $A$ is the amplitude such that $A = 1~{\rm{cm}}$, because it is the maximum distance. Let's suppose that $x_0 = 0.5~{\rm{cm}}$ is achieved at $t=t_1$ and $A$ is achieved at $t = t_2$. Then,
$$x_0 = A~\sin{~\omega t_1} \;(1)$$
$$A = A~\sin{~\omega t_2}$$ or
$$\sin{~\omega t_2} = 1$$
Hence, $\omega t_2 = \frac{\pi}{2} + 2k\pi$ with $k\in\mathbb{Z}$, for $k=0$, $t_2 = \frac{\pi}{2\omega}$. As $\omega = \frac{2\pi}{T}$, so $t_2 = \frac{T}{4}$. Since $\Delta t = t_2 - t_1 = 0.01~{\rm{s}}$,
$$t_1 = \frac{T}{4} - \Delta t \;(2)$$
Putting (2) into (1) and separating $T$, it is obtained
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<meta name="description" content="A plane capacitor is placed in a homogeneous magnetic field od induction B parallel to the plates. From point A, electrons enter perpendicularly to magnetic field direction. The tension (voltage) applied to the plates is V. What is the condition for which electrons pass through the capacitor?">
<meta name="description" content="A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?">
<meta property="og:title" content="A plane capacitor is placed in a homogeneous magnetic field od induction B parallel to the plates. From point A, electrons enter perpendicularly to magnetic field direction. The tension (voltage) applied to the plates is V. What is the condition for which electrons pass through the capacitor?">
<meta property="og:title" content="A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?">
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<meta property="og:description" content="A plane capacitor is placed in a homogeneous magnetic field od induction B parallel to the plates. From point A, electrons enter perpendicularly to magnetic field direction. The tension (voltage) applied to the plates is V. What is the condition for which electrons pass through the capacitor?">
<meta property="og:description" content="A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?">
<title>A plane capacitor is placed in a homogeneous magnetic field od induction B parallel to the plates. From point A, electrons enter perpendicularly to magnetic field direction. The tension (voltage) applied to the plates is V. What is the condition for which electrons pass through the capacitor?</title>
<title>A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?</title>
A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?
A weight, oscillating freely on a spring, has moved from a distance of 0.5 cm from its equilibrium position to the largest one, equal to 1 cm, for a time of 0.01 s. What is the period of its oscillations?
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
For a harmonic motion, variable position depends sinusoidally on time. In this case, we suppose that load is at equilibrium position at $t = 0$. So, sine function is more adequate to this situation without phase angle or null phase angle. So,
For a harmonic motion, variable position depends sinusoidally on time. In this case, we suppose that load is at equilibrium position at $t = 0$. So, sine function is more adequate to this situation without phase angle or null phase angle. So,
$$x(t) = A~\sin{~\omega t}$$
$$x(t) = A~\sin{~\omega t}$$
where $A$ is the amplitude such that $A = 1~{\rm{cm}}$, because it is the maximum distance. Let's suppose that $x_0 = 0.5~{\rm{cm}}$ is achieved at $t=t_1$ and $A$ is achieved at $t = t_2$. Then,
where $A$ is the amplitude such that $A = 1~{\rm{cm}}$, because it is the maximum distance. Let's suppose that $x_0 = 0.5~{\rm{cm}}$ is achieved at $t=t_1$ and $A$ is achieved at $t = t_2$. Then,
$$x_0 = A~\sin{~\omega t_1} \;(1)$$
$$x_0 = A~\sin{~\omega t_1} \;(1)$$
$$A = A~\sin{~\omega t_2}$$ or
$$A = A~\sin{~\omega t_2}$$ or
$$\sin{~\omega t_2} = 1$$
$$\sin{~\omega t_2} = 1$$
Hence, $\omega t_2 = \frac{\pi}{2} + 2k\pi$ with $k\in\mathbb{Z}$, for $k=0$, $t_2 = \frac{\pi}{2\omega}$. As $\omega = \frac{2\pi}{T}$, so $t_2 = \frac{T}{4}$. Since $\Delta t = t_2 - t_1 = 0.01~{\rm{s}}$,
Hence, $\omega t_2 = \frac{\pi}{2} + 2k\pi$ with $k\in\mathbb{Z}$, for $k=0$, $t_2 = \frac{\pi}{2\omega}$. As $\omega = \frac{2\pi}{T}$, so $t_2 = \frac{T}{4}$. Since $\Delta t = t_2 - t_1 = 0.01~{\rm{s}}$,
$$t_1 = \frac{T}{4} - \Delta t \;(2)$$
$$t_1 = \frac{T}{4} - \Delta t \;(2)$$
Putting (2) into (1) and separating $T$, it is obtained
Putting (2) into (1) and separating $T$, it is obtained
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
