Types of Acceleration in Rotation: Videos & Practice Problems

In rotational motion, four types of acceleration are crucial: centripetal acceleration (A_C), tangential acceleration (A_T), total acceleration (A), and angular acceleration (α). A_C maintains circular paths, while A_T and α indicate changes in speed. The relationship A = √(A_T² + A_C²) combines these accelerations. For example, in a carousel, if the angular velocity is constant, A_T and α are zero, simplifying total acceleration to A_C. Understanding these concepts is essential for solving rotation problems effectively.

Types of Acceleration in Rotation

In rotational motion, understanding the different types of acceleration is crucial for solving problems effectively. There are four primary types of acceleration, each with its own significance and sometimes referred to by multiple names. The first type is centripetal acceleration (denoted as \( a_c \) or \( a_{rad} \)), which is responsible for keeping an object moving in a circular path. It is calculated using the formula:

\( a_c = \frac{v^2}{r} \)

where \( v \) is the tangential velocity and \( r \) is the radius of the circular path. This can also be expressed in terms of angular velocity (\( \omega \)) as:

\( a_c = r \omega^2 \)

The second type is tangential acceleration (denoted as \( a_t \)), which is sometimes called linear acceleration. This acceleration occurs when an object is speeding up or slowing down along its circular path. The relationship between tangential acceleration and angular acceleration (\( \alpha \)) is given by:

\( a_t = r \alpha \)

Next, we have total acceleration (simply referred to as \( a \)), which combines both centripetal and tangential accelerations. The total acceleration can be calculated using vector addition, specifically the Pythagorean theorem:

\( a = \sqrt{a_t^2 + a_c^2} \)

Finally, angular acceleration (denoted as \( \alpha \)) measures how quickly the angular velocity of an object is changing and is expressed in radians per second squared. It is important to note that if an object is moving at a constant speed in a circular path, both tangential acceleration and angular acceleration will be zero.

In practical applications, such as analyzing a carousel, one can determine the tangential velocity using the relationship:

\( v_t = r \omega \)

where \( \omega \) can be derived from the period of rotation. For example, if a carousel has a radius of 10 meters and completes one cycle every 75 seconds, the angular velocity can be calculated as:

\( \omega = \frac{2\pi}{T} \)

Substituting the period \( T \) gives the angular velocity, which can then be used to find the tangential velocity. If the object is not accelerating, the tangential acceleration and angular acceleration will be zero, simplifying the calculations for total acceleration to just the centripetal acceleration.

Understanding these concepts and their interrelations is essential for solving rotational motion problems effectively, allowing for a clear approach to analyzing forces and accelerations in circular motion.

Accelerations of boy on carousel

In this scenario, we analyze a carousel with a radius of 16 meters that accelerates from rest. The initial angular velocity (\(\omega_{\text{initial}}\)) is 0, and the angular acceleration (\(\alpha\)) is 0.05 radians per second squared. A buoy stands at the edge of the carousel, maintaining a distance (\(r\)) of 16 meters from the center. We aim to determine the tangential velocity after 10 seconds.

The tangential velocity (\(v_t\)) of an object on the edge of a rotating disk is calculated using the formula:

\(v_t = r \cdot \omega\)

where \(r\) is the radius and \(\omega\) is the angular speed. To find \(\omega\) after 10 seconds, we use the angular motion equation that does not involve angular displacement:

\(\omega_{\text{final}} = \omega_{\text{initial}} + \alpha \cdot t\)

Substituting the known values:

\(\omega_{\text{final}} = 0 + 0.05 \cdot 10 = 0.5 \text{ radians per second}\)

Now, substituting \(\omega\) back into the tangential velocity equation:

\(v_t = 16 \cdot 0.5 = 8 \text{ meters per second}\)

Next, we calculate the tangential acceleration (\(a_t\)), which is given by:

\(a_t = r \cdot \alpha\)

Substituting the values:

\(a_t = 16 \cdot 0.05 = 0.8 \text{ meters per second squared}\)

For the radial (centripetal) acceleration (\(a_{\text{rad}}\)), we can use the formula:

\(a_{\text{rad}} = \frac{v^2}{r}\) or \(a_{\text{rad}} = r \cdot \omega^2\). Using the first formula:

\(a_{\text{rad}} = \frac{8^2}{16} = 4 \text{ meters per second squared}\)

The angular acceleration (\(\alpha\)) remains constant at 0.05 radians per second squared, as previously established.

Finally, to find the total linear acceleration (\(a\)), we combine the tangential and radial accelerations using the Pythagorean theorem:

\(a = \sqrt{a_t^2 + a_{\text{rad}}^2}\)

Substituting the values:

\(a = \sqrt{(0.8)^2 + (4)^2} = \sqrt{0.64 + 16} = \sqrt{16.64} \approx 4.1 \text{ meters per second squared}\)

In summary, after 10 seconds, the tangential velocity is 8 meters per second, the tangential acceleration is 0.8 meters per second squared, the radial acceleration is 4 meters per second squared, the angular acceleration is 0.05 radians per second squared, and the total linear acceleration is approximately 4.1 meters per second squared.

A large disc of radius 10 m initially at rest takes 200 full revolutions to reach 30 RPM. Calculate the total linear acceleration of a point at half way between the disc's center and its edge, once the disc reaches 30 RPM. (You may assume it continues accelerating past that point)

• A. 15.7 m/s²
• B. 24.7 m/s²
• C. 49.3 m/s²
• D. 98.7 m/s²

Answer: (C)

An object of negligible size moves in a circular path of radius 20 m with 90 RPM. Find its radial acceleration.

• A. 0 m/s²
• B. 444 m/s²
• C. 592 m/s²
• D. 1776 m/s²

Answer: (D)

Here’s what students ask on this topic:

What are the different types of acceleration in rotational motion?

In rotational motion, there are four key types of acceleration: centripetal acceleration (AC), tangential acceleration (AT), total acceleration (A), and angular acceleration (α). Centripetal acceleration (AC) maintains the circular path of an object and is directed towards the center of the circle. Tangential acceleration (AT) indicates changes in the speed along the tangent to the circular path. Total acceleration (A) is the vector sum of centripetal and tangential accelerations, given by the equation $\sqrt{a^2+a^2}$. Angular acceleration (α) describes the rate of change of angular velocity and is measured in radians per second squared.

How do you calculate centripetal acceleration in rotational motion?

Centripetal acceleration (AC) in rotational motion can be calculated using the formula $ac=\frac{v^2}{r}$, where $v$ is the tangential velocity and $r$ is the radius of the circular path. Alternatively, it can also be expressed as $ac=r{\omega }^2$, where $\omega$ is the angular velocity.

What is the relationship between tangential acceleration and angular acceleration?

Tangential acceleration (AT) and angular acceleration (α) are related by the equation $at=r\alpha$, where $r$ is the radius of the circular path. This relationship shows that tangential acceleration is directly proportional to both the radius and the angular acceleration. If the angular acceleration is zero, the tangential acceleration will also be zero, indicating no change in the tangential speed of the object.

How do you find the total acceleration in rotational motion?

Total acceleration (A) in rotational motion is the vector sum of centripetal acceleration (AC) and tangential acceleration (AT). It can be calculated using the Pythagorean theorem: $A=\sqrt{a^{\mathrm{t}}+a^{\mathrm{c}}}$. This equation combines the magnitudes of the tangential and centripetal accelerations to give the total acceleration experienced by the object in rotational motion.

What is the significance of angular acceleration in rotational motion?

Angular acceleration (α) is significant in rotational motion as it describes the rate of change of angular velocity. It indicates how quickly an object is speeding up or slowing down its rotation. Angular acceleration is measured in radians per second squared and is crucial for understanding the dynamics of rotating systems. It helps in determining the tangential acceleration and, consequently, the total acceleration of the object in rotational motion.