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Magazine Chapter 7: Problem 27
The pendulum oscillates about the \(x\) -axis according to \(\theta=\frac{\pi}{6} \sin 3 \pi t\) radians, where \(t\) is the time in seconds. Simultaneously, the shaft \(O A\) revolves about the vertical \(z\) -axis at the constant rate \(\omega_{z}=2 \pi \mathrm{rad} / \mathrm{sec}\) Determine the velocity \(\mathbf{v}\) and acceleration a of the center \(B\) of the pendulum as well as its angular acceleration \(\alpha\) for the instant when \(t=0\)
Short Answer
Expert verified
At \(t=0\), \(\theta = 0\), \(\dot{\theta} = \frac{\pi^2}{2}\), \(\alpha = 0\); velocity and acceleration depend on both oscillation and rotation effects.
Step by step solution
01
Define the problem and given values
We need to determine the velocity \(\mathbf{v}\), acceleration \(a\), and angular acceleration \(\alpha\) of the pendulum at \(t=0\). The pendulum's angular position \(\theta\) is given by \(\theta = \frac{\pi}{6} \sin 3 \pi t \), and the shaft rotates with \(\omega_z = 2\pi\) rad/sec.
02
Calculate angular position \(\theta\) at \(t=0\)
Substitute \(t=0\) into the given angular position function:\[\theta = \frac{\pi}{6} \sin(3\pi \cdot 0) = 0\]This means the pendulum is in its equilibrium position when \(t=0\).
03
Calculate angular velocity \(\dot{\theta}\) at \(t=0\)
Differentiate \(\theta(t)\) with respect to time \(t\):\[\dot{\theta} = \frac{d}{dt}\left( \frac{\pi}{6} \sin 3\pi t \right) = \frac{\pi}{6} \cdot 3\pi \cos(3\pi t)\]At \(t=0\):\[\dot{\theta} = \frac{\pi}{6} \cdot 3\pi \cdot \cos(0) = \frac{3\pi^2}{6} = \frac{\pi^2}{2}\]
04
Calculate angular acceleration \(\alpha\) at \(t=0\)
Differentiate \(\dot{\theta}\) with respect to time \(t\) to find \(\alpha\):\[\alpha = \frac{d}{dt} \left( \frac{3\pi^2}{6} \cos(3\pi t) \right) = -\frac{3\pi^3}{6} \sin(3\pi t)\]At \(t=0\):\[\alpha = -\frac{3\pi^3}{6} \cdot \sin(0) = 0\]
05
Consider shaft rotation and compute velocity \(\mathbf{v}\) of pendulum center
The velocity due to shaft rotation is tangent to the circle described by the revolving system. Since the angular velocity about the vertical \(z\)-axis is constant at \(\omega_z = 2\pi\), the linear velocity of point \(B\) due to the rotation about the \(z\)-axis is also essential. Without the radius of pendulum, compute relative contributions, assuming length \(L\):\[v_B = L \cdot \sqrt{(\dot{\theta})^2 + \omega_z^2} = L \cdot \sqrt{\left( \frac{\pi^2}{2} \right)^2 + (2\pi)^2}\]
06
Calculate the total acceleration \(a\) of point \(B\)
The acceleration consists of tangential and centripetal components due to the angular velocities and angular acceleration:- Tangential acceleration from \(\theta\) is \(a_t = L \cdot \alpha\),- Centripetal acceleration from \(\theta\) is \(a_c = L \cdot \dot{\theta}^2\),- Additional centripetal acceleration due to shaft rotation is \(a_{cz} = L \cdot \omega_z^2\).Total acceleration:\[a_B = \sqrt{a_t^2 + a_c^2 + a_{cz}^2}\]
07
Conclusion
At \(t=0\), the angular position \(\theta = 0\), the angular velocity of the pendulum \(\dot{\theta} = \frac{\pi^2}{2}\), angular acceleration \(\alpha = 0\), and the velocity and acceleration components incorporate changes from both \(\theta\) oscillation and shaft rotation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angular Motion
Angular motion is the rotational movement of an object around a central point or axis. In the context of pendulum dynamics, this involves the back-and-forth swinging that occurs when a pendulum rotates about its pivot point. The pendulum's motion can be described using various terms like angular position, velocity, and acceleration. Each of these terms captures different aspects of the pendulum's movement.
Angular position tells us about the orientation of the pendulum at a given time. It is usually measured in radians. In our problem, the angular position is given as \( \theta = \frac{\pi}{6} \sin(3\pi t) \), which changes over time as the pendulum swings.
To analyze angular motion, it's important to understand how the oscillating angle \( \theta \) impacts the motion dynamics of the pendulum. The further the angular position is from zero, the further the pendulum is from its equilibrium position, and the more potential energy it possesses.
Angular position tells us about the orientation of the pendulum at a given time. It is usually measured in radians. In our problem, the angular position is given as \( \theta = \frac{\pi}{6} \sin(3\pi t) \), which changes over time as the pendulum swings.
To analyze angular motion, it's important to understand how the oscillating angle \( \theta \) impacts the motion dynamics of the pendulum. The further the angular position is from zero, the further the pendulum is from its equilibrium position, and the more potential energy it possesses.
Oscillation
Oscillation refers to any repetitive variation, typically in time, of a quantity or position around a central value. For a pendulum, oscillation describes the repetitive swinging motion around its rest position. A pendulum is generally defined by its periodic nature, as it swings back and forth in a regular time interval.
Several key variables define an oscillation:
Understanding the dynamics of oscillation is crucial to predicting how forces like gravity and inertia will influence the pendulum's movement at any given time.
Several key variables define an oscillation:
- The amplitude, which is the maximum extent of the pendulum's swing from the central (or rest) position.
- The frequency, which is the number of oscillations made per unit of time.
- The period, which is the time taken for one complete cycle of oscillation.
Understanding the dynamics of oscillation is crucial to predicting how forces like gravity and inertia will influence the pendulum's movement at any given time.
Angular Velocity
Angular velocity refers to how fast an angle changes with time. It's a measure of the rate of rotation around an axis. For the pendulum, it's essential to distinguish between the angular velocity of the pendulum's swing and the shaft's constant rotation.
In our example, the pendulum's angular velocity at \( t = 0 \) was calculated by differentiating the angle \( \theta \) with respect to time to be \( \dot{\theta} = \frac{\pi^2}{2} \) rad/s. This represents how fast the pendulum swings at that instant.
On the other hand, the shaft's rotation around the vertical \( z \)-axis is constant at \( \omega_z = 2\pi \) rad/s. This constant rotational speed must be considered when calculating the pendulum's total velocity, as it contributes significantly to how fast the pendulum's center point \( B \) moves in space.
Angular velocity is an essential concept as it provides insight into how quick an object is spinning and influences the linear velocity and kinetic energy of the system.
In our example, the pendulum's angular velocity at \( t = 0 \) was calculated by differentiating the angle \( \theta \) with respect to time to be \( \dot{\theta} = \frac{\pi^2}{2} \) rad/s. This represents how fast the pendulum swings at that instant.
On the other hand, the shaft's rotation around the vertical \( z \)-axis is constant at \( \omega_z = 2\pi \) rad/s. This constant rotational speed must be considered when calculating the pendulum's total velocity, as it contributes significantly to how fast the pendulum's center point \( B \) moves in space.
Angular velocity is an essential concept as it provides insight into how quick an object is spinning and influences the linear velocity and kinetic energy of the system.
Angular Acceleration
Angular acceleration is the rate at which the angular velocity changes with time. It's an important factor when analyzing how forces, such as torque, impact the rotational motion of an object. In the pendulum example, angular acceleration provides insight into how quickly the pendulum's swinging speed changes.
In step 4 of our solution, angular acceleration \( \alpha \) was determined by further differentiating the angular velocity \( \dot{\theta} \). When evaluated at \( t = 0 \), \( \alpha \) was calculated to be zero, indicating that at this particular moment, the pendulum isn't speeding up or slowing down rotationally.
Generally, angular acceleration can result from changes in the driving forces or interactions from external movements, such as the constant shaft rotation in this example. It's crucial for understanding how the dynamics of movement translate from rotational motion to linear acceleration in connected components of the system.
By understanding angular acceleration, students can predict how changes in forces will affect the system's motion over time.
In step 4 of our solution, angular acceleration \( \alpha \) was determined by further differentiating the angular velocity \( \dot{\theta} \). When evaluated at \( t = 0 \), \( \alpha \) was calculated to be zero, indicating that at this particular moment, the pendulum isn't speeding up or slowing down rotationally.
Generally, angular acceleration can result from changes in the driving forces or interactions from external movements, such as the constant shaft rotation in this example. It's crucial for understanding how the dynamics of movement translate from rotational motion to linear acceleration in connected components of the system.
By understanding angular acceleration, students can predict how changes in forces will affect the system's motion over time.
