In a department store toy display, a small disk (disk 1) of radius \(0.100 \mathrm{~m}\) is driven by a motor and turns a larger disk (disk 2) of radius \(0.500 \mathrm{~m}\). Disk 2 , in turn, drives disk 3 , whose radius is \(1.00 \mathrm{~m}\). The three disks are in contact and there is no slipping. Disk 3 is observed to sweep through one complete revolution every \(30.0 \mathrm{~s}\) a) What is the angular speed of disk \(3 ?\) b) What is the ratio of the tangential velocities of the rims of the three disks? c) What is the angular speed of disks 1 and \(2 ?\) d) If the motor malfunctions, resulting in an angular acceleration of \(0.100 \mathrm{rad} / \mathrm{s}^{2}\) for disk 1 , what are disks 2 and 3's angular accelerations?

When an object moves in a circular path, tangential velocity refers to the linear velocity of any point on the object. It's the rate at which the point is covering distance along the circular path and is tangent to the rotation circle itself, pointing in the direction of motion.

Mathematically, tangential velocity, denoted as V, is found using the equation:
\[V = r\omega\]
where r is the radius of the circular path, and \(\omega\) is the angular speed of the object. In a system where multiple objects are connected, like the toy disks in the exercise, we can assess the relative motion between them through their tangential velocities. If there's no slipping, the tangential velocity at the edge of one disk will be equal to the tangential velocity at the edge of the other where they contact each other. This ensures a common speed at the point of contact, which is crucial for predicting the behavior of connected rotating systems.

Angular acceleration is the rate at which angular speed changes with time. In scenarios where discs or wheels are being spun up or down by forces or torques, we use angular acceleration to describe how quickly they are picking up or losing speed in their rotation.

The equation for angular acceleration, denoted by \(\alpha\), is:
\[\alpha = \frac{\Delta\omega}{\Delta t}\]
where \(\Delta\omega\) signifies the change in angular speed and \(\Delta t\) is the change in time. This concept extends the understanding of rotational motion into a dynamic realm where speeds aren't constant, as illustrated when the problem discusses the malfunction of the motor leading to an angular acceleration on disk 1. The direct proportionality between the radii of the disks and their angular accelerations is a critical point, mirroring the no-slip condition applied to angular speeds.

Rotational motion is the movement of a body about an axis. Everything that spins, from wheels to planets, exhibits this type of motion. The central concepts in rotational motion are angular speed and angular acceleration, which together describe how quickly an object rotates (\(\omega\)) and how that speed changes over time (\(\alpha\)).

In the toy disk example, each disk exhibits rotational motion. By understanding the relationships between the radii of the disks and the no-slip condition, we're able to determine that the ratio of their rotational speeds is inverse to the ratio of their radii. This is fundamental in systems with gears or interconnected wheels, ensuring that the motion is transmitted effectively from one object to another without energy losses due to slippage.

The no-slip condition in mechanics refers to the assumption that two rolling bodies in contact move without slipping against each other. This means that at the point of contact, both bodies have the same tangential velocity.

When this condition is applied to a set of rotating disks or gears, it asserts that the speed at which the edges of two contacting bodies move relative to their centers is the same. This can be expressed with the formula:
\[r_1\omega_1 = r_2\omega_2\]
where r represents the radius, and \(\omega\) the angular speed of the rotating objects. The condition ensures a fixed transmission ratio, which is vital in mechanisms where exact timing between the rotation of different parts is essential, such as in clocks or the toy display described in the exercise. When considering angular acceleration, the analog condition is applied to maintain the proportional relationship across different radii.