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Chapter 12: Problem 160

The box slides down the helical ramp such that \(r=0.5 \mathrm{~m}, \theta=\left(0.5 t^{3}\right) \mathrm{rad},\) and \(z=\left(2-0.2 t^{2}\right) \mathrm{m},\) where \(t\) is in seconds. Determine the magnitudes of the velocity and acceleration of the box at the instant \(\theta=2 \pi \mathrm{rad}\).

Short Answer

Expert verified
The magnitudes of the velocity and acceleration of the box when \( \theta = 2 \pi rad \) are approximately 5.325 m/s and 7.114 m/s^2, respectively.

Step by step solution

01

Identify the given functions of motion

The problem gives three functions: \( r = 0.5 m \) which is constant, \( \theta = 0.5 t^3 \) rad, and \( z= 2 - 0.2 t^2 \) m. These represent the radial, angular and vertical positions of the box on the helical ramp. We need to differentiate these to find the velocity and acceleration.
02

Differentiate to find velocity

Differentiate each function with respect to time to find their rates of changes, which are their respective velocities (linear or angular). The derivative of the constant function \( r = 0.5 m \) is \( \dot{r} = 0 m/s \). The derivative of \( \theta = 0.5 t^3 \) rad is \( \dot{\theta} = 1.5 t^2 rad/s \), and the derivative of \( z = 2 - 0.2 t^2 \) is \( \dot{z} = -0.4 t m/s \). These give us the components of the velocity.
03

Differentiate again to find acceleration

Differentiating the functions of velocity will give us the acceleration. The only function in step 2 that depends on time is \( \dot{\theta} = 1.5 t^2 \) rad/s. Its derivative, \( \ddot{\theta} = 3t rad/s^2 \), is the angular acceleration. The radial and vertical accelerations, \( \ddot{r} \) and \( \ddot{z} \), are zero because we found in step 2 that their velocities didn't change with time.
04

Substitute the given moment \(\theta = 2 \pi rad \) into the velocity and acceleration functions

Since \( \theta = 0.5 t^3 \) rad, when \( \theta = 2 \pi rad \), solving for \( t \) gives \( t = (4 \pi / 0.5)^{1/3} = ~2.5198 s \). Substituting this into the velocity components gives \( \dot{r} = 0 m/s \), \( \dot{\theta} = 1.5 (2.5198)^2 = ~9.534 rad/s \), and \( \dot{z} = -0.4 (2.5198) = -1.008 m/s \). Thus, the magnitude of the velocity \( V = \sqrt{\dot{r}^2 + (\dot{\theta} r)^2 + \dot{z}^2} = 5.325 m/s \). Similarly, the angular acceleration \( \ddot{\theta} = 3 (2.5198) = ~7.559 rad/s^2 \), so the magnitude of the acceleration \( A = \sqrt{\ddot{r}^2 + (\ddot{\theta} r + \dot{\theta}^2 r)^2 + \ddot{z}^2} = ~7.114 m/s^2 \).
05

Conclude the problem

The magnitudes of the velocity and acceleration when the box has made one complete revolution (i.e. \( \theta = 2 \pi rad \)) are ~5.325 m/s and ~7.114 m/s^2, respectively.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinematics
Kinematics is a branch of mechanics that describes the motion of objects without analyzing the forces causing them. It involves parameters such as position, velocity, and acceleration. In our exercise, we have three key components of kinematics to consider:

  • **Radial motion**: This component gives us an idea of how far a point is from the axis of rotation. Here the radius \( r = 0.5 \) m is constant, indicating no radial motion.

  • **Angular motion**: Represented by the angle \( \theta = 0.5 t^3 \) rad, which changes with time. The angle shows the rotational aspect of the helical path.

  • **Vertical motion**: Given by \( z = 2 - 0.2 t^2 \) m, this indicates how the height of the object changes as it moves along the helix.

Understanding these components helps to analyze how an object travels in a path, which is crucial in many engineering applications like designing ramps or slides.
Helical motion
Helical motion occurs when an object travels in a path that spirals around an axis, combining circular (angular) motion with linear motion parallel to the axis. This can look like a spring way or a corkscrew path. In this problem, the box sliding down the ramp showcases helical motion.

Helical motion has unique characteristics:
  • **Combination of motions**: It combines rotational (angular) and vertical (linear) motion. Therefore, an object in helical motion has both a rotational velocity and a linear velocity.

  • **Representation**: It can be broken down into three parameters: the radius of the helix \( r \), the angle \( \theta \) describing the circular motion, and the height \( z \) which changes vertically.

  • **Real-world applications**: Helical motion is common in engineering devices like screws, drills, and conveyors, where precise control of motion along both dimensions is required.
Velocity and acceleration analysis
Velocity and acceleration are essential for understanding motion dynamics. To determine these in helical motion, we need to differentiate the object鈥檚 position with respect to time.

**Velocity** is the rate of change of position. There are different components:
  • **Radial velocity**: Given the constant radial distance \( r = 0.5 \), radial velocity is \( \dot{r} = 0 \) m/s.

  • **Angular velocity**: Derived from \( \theta = 0.5 t^3 \), differentiating gives \( \dot{\theta} = 1.5 t^2 \) rad/s, representing rotational speed.

  • **Vertical velocity**: For \( z = 2 - 0.2t^2 \), differentiating results in \( \dot{z} = -0.4t \) m/s, showing vertical displacement per time unit.

**Acceleration** is the rate of change of velocity. It also has components:
  • **Angular acceleration**: From \( \dot{\theta} \'s \) derivative, we get \( \ddot{\theta} = 3t \) rad/s虏.

  • **Vertical acceleration**: Vertical velocity \( \dot{z} = -0.4 t \) means vertical acceleration is zero as \( t \) is constant in this context.

In these calculations, substituting the moment when \( \theta = 2 \pi \) rad allows us to find the magnitudes by combining all components geometrically, as done in the original solution. Understanding these analyses is critical in predicting motion effects accurately for engineering structures and mechanisms.

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