Question

A rigid body, starting at rest, rotates about a fixed axis with constant angular acceleration \(\alpha\). Consider a particle a distance \(r\) from the axis. Express \((a)\) the radial acceleration and \((b)\) the tangential acceleration of this particle in terms of \(\alpha, r\), and the time \(t .(c)\) If the resultant acceleration of the particle at some instant makes an angle of \(57.0^{\circ}\) with the tangential acceleration, through what total angle has the body rotated from \(t=0\) to that instant?

Short answer

The total angle through which the body has rotated from \(t = 0\) to the given instant will be obtained from calculation in Step 4 using the time obtained from Step 3.

Step-by-step solution

Finding Radial Acceleration

The radial acceleration of a particle in rotational motion, sometimes called centripetal acceleration, can be expressed in terms of angular velocity (\(w\)) and the radial distance from the rotation axis (\(r\)). However, the problem specifies that the body starts from rest and then accelerates with a constant angular acceleration (\(α\)). Thus, the radial acceleration (\(a_r\)) can be found using the formula: \(a_{r} = \omega^{2}r\), and knowing that \(ω = ατ\), we have \(a_{r} = α^{2}t^{2}r\).

Finding Tangential Acceleration

The tangential acceleration (\(a_t\)) of a particle in rotational motion is given by the change in velocity with time. Because the particle starts from rest, the tangential acceleration can be expressed using the equation: \(a_{t}=αr\).

Calculating Resultant Acceleration

The resultant acceleration (\(a\)) is the vector sum of the radial (\(a_r\)) and tangential (\(a_t\)) accelerations. This resultant acceleration makes an angle of \(57.0^{\circ}\) with the tangential acceleration. Therefore, we can express the resultant acceleration using trigonometry as: \(a = a_{t}/cos(57.0^{\circ})\), which can be simplified to \(a = αr/sec(57.0^{\circ})\). Remember that this resultant acceleration should also be valid when squared and added to radial acceleration according to pythagorean theorem, i.e., \(a^{2}=a_{r}^{2}+a_{t}^{2}\). We may equate this equation to our previous expression and solve for \(t\).

Calculating Rotational Angle

We know that angular displacement (\(θ\)) is given by \(θ = ωt + 0.5αt^{2}\). As the rigid body starts from rest, ω equals 0 and the equation simplifies to \(θ = 0.5αt^{2}\). Plug in the value of \(t\) obtained from the previous step to get the total angle the body has rotated from \(t = 0\) to that instant.

Key concepts

Angular Acceleration

Angular acceleration is a fundamental concept in rotational motion. It measures how quickly the angular velocity of an object is changing. For an object rotating around a fixed axis, it is denoted by the Greek letter alpha ( \(\alpha\)), and is usually expressed in radians per second squared. Angular acceleration plays a crucial role in determining how quickly an object speeds up or slows down in its rotational path.

• When a rigid body starts from rest, the initial angular velocity is zero. If it gains angular acceleration, it means that over time, the angular velocity will increase.
• A constant angular acceleration signifies that the rate of change of angular velocity does not vary over time.
• This property is significant as it helps in predicting the future rotational motion of an object when the angular acceleration and initial conditions are known.

Angular acceleration directly influences both the radial and tangential components of acceleration for any particle located at a distance from the axis of rotation.

Radial Acceleration

Radial acceleration, often termed centripetal acceleration, is an important component for objects moving in circular paths. It represents the acceleration directed towards the rotation center (axis). This type of acceleration is crucial for maintaining the circular motion of a particle or object.

For a particle at a distance ( \(r\)) from the rotation axis, the radial acceleration ( \(a_r\)) depends on the angular velocity ( \(\omega\)). Mathematically, it can be expressed as:\[a_r = \omega^2 r\]

Given the particle starts from rest and accelerates constantly, \(\omega\) can be substituted using the relationship \(\omega = \alpha t\):\[a_r = \alpha^2 t^2 r\]

• This shows how radial acceleration increases with time as both \(\alpha\) and \(t\) play crucial roles.
• This component ensures that the object remains in its rotational path by pulling it towards the center.

Tangential Acceleration

Tangential acceleration is the component of acceleration that acts along the edge of a rotating circle. It dictates how quickly an object can change its speed along the circular path. For a particle at a radius \(r\), the tangential acceleration ( \(a_t\)) due to a given angular acceleration ( \(\alpha\)) is calculated as follows:\[a_t = \alpha r\]

• The tangential acceleration is directly proportional to both the angular acceleration and the radius.
• It affects how fast the velocity along the tangent at a certain point on the rotating circle changes.
• An increase in \(\alpha\) results in a proportional increase in tangential acceleration, impacting the linear speed along the circle.

This component contributes to the change in the speed of rotation and is essential for understanding the dynamics of rotating systems.

Rotational Angle

The rotational angle describes the degree to which a body has rotated around an axis. It is denoted by the Greek letter theta ( \(\theta\)) and is usually measured in radians. This measure allows us to track how much an object has turned between its initial and final positions over time.

For a rigid body starting from rest under constant angular acceleration, the rotational angle can be determined using:\[\theta = \frac{1}{2} \alpha t^2\]

• The formula highlights that the rotational angle is determined by both the magnitude of the angular acceleration and the elapsed time.
• As the body accelerates over time, the rotational angle increases, signifying greater rotations about the axis.
• This relationship is particularly important for understanding how rotational motion evolves over time, especially in physics and engineering applications.

By calculating the total angle rotated, we can understand the extent of rotation undergone by a body from rest to any given time under constant acceleration.