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<title>A heavy cart rolls with acceleration a on an inclined plane forming an angle \alpha with the horizon. Find the period of oscillation of a pendulum of length l mounted on the cart.</title>
$3.2.16.$ A heavy cart rolls with acceleration $a$ on an inclined plane forming an angle $\alpha$ with the horizon. Find the period of oscillation of a pendulum of length $l$ mounted on the cart.
Acceleration vector $\vec{g}^*$ in an inertial frame of reference
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<p>
The period of oscillation of a pendulum on an inclined cart can be found using the well-known formula for the period of oscillation of a mathematical pendulum, by switching to the inertial frame of reference of the cart
$$ T=2\pi\sqrt{\frac{l}{g^*}}\quad(1) $$
@@ -83,19 +83,19 @@
Where $\vec{g}^*$ is the acceleration of the inertial reference frame
$$\vec{g}^*=\vec{g}-\vec{a}$$
Using the cosine law, we find the value of the modulus of the vector $\vec{g}^*$
−
$$ (g^{*})^2=a^2+g^2-2agcos\alpha$$
+
$$ (g^{*})^2=a^2+g^2-2ag\cos\alpha$$
Where
−
$$\boxed{g^*=\sqrt{a^2+g^2-2agcos\alpha}}$$
+
$$\boxed{g^*=\sqrt{a^2+g^2-2ag\cos\alpha}}$$
We substitute $(1)$ into the expression and find the period of oscillation of the pendulum
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<title>A heavy cart rolls with acceleration a on an inclined plane forming an angle \alpha with the horizon. Find the period of oscillation of a pendulum of length l mounted on the cart.</title>
<title>A heavy cart rolls with acceleration a on an inclined plane forming an angle \alpha with the horizon. Find the period of oscillation of a pendulum of length l mounted on the cart.</title>
$3.2.16.$ A heavy cart rolls with acceleration $a$ on an inclined plane forming an angle $\alpha$ with the horizon. Find the period of oscillation of a pendulum of length $l$ mounted on the cart.
$3.2.16.$ A heavy cart rolls with acceleration $a$ on an inclined plane forming an angle $\alpha$ with the horizon. Find the period of oscillation of a pendulum of length $l$ mounted on the cart.
Acceleration vector $\vec{g}^*$ in an inertial frame of reference
Acceleration vector $\vec{g}^*$ in an inertial frame of reference
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
The period of oscillation of a pendulum on an inclined cart can be found using the well-known formula for the period of oscillation of a mathematical pendulum, by switching to the inertial frame of reference of the cart
The period of oscillation of a pendulum on an inclined cart can be found using the well-known formula for the period of oscillation of a mathematical pendulum, by switching to the inertial frame of reference of the cart
$$ T=2\pi\sqrt{\frac{l}{g^*}}\quad(1) $$
$$ T=2\pi\sqrt{\frac{l}{g^*}}\quad(1) $$
@@ -83,19 +83,19 @@
Where $\vec{g}^*$ is the acceleration of the inertial reference frame
Where $\vec{g}^*$ is the acceleration of the inertial reference frame
$$\vec{g}^*=\vec{g}-\vec{a}$$
$$\vec{g}^*=\vec{g}-\vec{a}$$
Using the cosine law, we find the value of the modulus of the vector $\vec{g}^*$
Using the cosine law, we find the value of the modulus of the vector $\vec{g}^*$
$$ (g^{*})^2=a^2+g^2-2agcos\alpha$$
$$ (g^{*})^2=a^2+g^2-2ag\cos\alpha$$
Where
Where
$$\boxed{g^*=\sqrt{a^2+g^2-2agcos\alpha}}$$
$$\boxed{g^*=\sqrt{a^2+g^2-2ag\cos\alpha}}$$
We substitute $(1)$ into the expression and find the period of oscillation of the pendulum
We substitute $(1)$ into the expression and find the period of oscillation of the pendulum
<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>
