A uniform disk with radius R = 0.400 m and mass 30.0 kg...

A uniform disk with radius $R =$ 0.400 m and mass 30.0 kg rotates in a horizontal plane on a frictionless vertical axle that passes through the center of the disk. The angle through which the disk has turned varies with time according to $\theta(t) =$ (1.10 rad/s)$t +$ (6.30 rad/s$^2)t^2$. What is the resultant linear acceleration of a point on the rim of the disk at the instant when the disk has turned through 0.100 rev?

Key Concepts

Rotational Kinematics

Rotational kinematics deals with the motion of bodies that rotate about an axis. It includes concepts analogous to linear motion such as angular displacement, angular velocity, and angular acceleration. In problems involving rotation, these quantities and their relationships help describe how the position of a rotating body changes over time and how the rate of rotation changes, which is central to determining various effects like linear acceleration at points on the rotating object.

Angular Acceleration

Angular acceleration is the rate at which the angular velocity of a rotating body changes with time. It indicates how quickly the rotation speeds up or slows down. In mathematical terms, it is the second derivative of the angular displacement with respect to time. This concept is crucial when the rotation is not uniform, as it provides the basis for calculating the tangential acceleration experienced by points on a rotating body.

Tangential Acceleration

Tangential acceleration is the linear acceleration experienced by a point on the rim of a rotating object due to the change in the magnitude of the angular velocity. It is calculated by multiplying the radius of the rotation by the angular acceleration. This component of acceleration points along the tangent to the circular path and is responsible for changes in the speed of the point’s motion along that path.

Centripetal (Radial) Acceleration

Centripetal acceleration is the acceleration directed towards the center of a circular path that keeps a rotating object moving along that curve. It arises from the change in direction of the velocity vector, even when the speed remains constant. The magnitude of centripetal acceleration is given by the square of the angular velocity times the radius. Together with tangential acceleration, it determines the net acceleration experienced by a point on the rim of a rotating body.

Unit Conversion (Revolutions to Radians)

Converting between revolutions and radians is fundamental when working with rotational quantities since the standard unit of angular measure in physics is the radian. One complete revolution is equal to 2? radians. This conversion is critical when the given angular displacement is provided in revolutions and calculations require the angle to be in radians.

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