The second object could have its mass close to the central axis. The first object might be a heavy ring supported by struts on an axle like a flywheel. Imagine two objects of the same mass with different distribution of that mass. The quantity mr 2 is defined as moment of inertia of a point mass about the center of rotation. Substitute Newton's second law into the definition for torque with θ of 90 degrees (a right angle between F and r) and use the relationship between linear acceleration and tangential angular acceleration to obtain t = r F = rma = mr 2 ( a/ r) = mr 2α. This physical quantity, torque, is t = r × F sin θ, where F is the force applied, r is the distance from the point of application to the center of the rotation, and θ is the angle from r to F. It is intuitive that the magnitude of the force applied and the distance from the point of application to the hinge affect the tendency of the door to rotate. It is easier to open a door by pushing on the edge farthest from the hinges than by pushing in the middle. The radial component of the linear acceleration is a r = v 2/ r = ω 2 r. The direction is the same as the velocity vector. This component of the acceleration is tangential to the point of rotation and represents the changing speed of the object. The average forward acceleration of the wheel is a T = r(ω f − ω o)/ t = rα. The direction of the velocity is tangent to the path of the point of rotation. In this case, the average forward speed of the wheel is v = d/ t = ( rθ)/ t = rω, where r is the distance from the center of rotation to the point of the calculated velocity. The forward displacement of the wheel is equal to the linear displacement of a point fixed on the rim. The kinematics equations for rotational motion at constant angular acceleration areĬonsider a wheel rolling without slipping in a straight line. The angular acceleration (α, Greek letter alpha) has the same form as the linear quantityĪnd is measured in radians/second/second or rad/s 2. The average angular velocity (ω, Greek letter omega), measured in radians per second, is The angular displacement of a rotating wheel is the angle between the radius at the beginning and the end of a given time interval. Many of the equations for the mechanics of rotating objects are similar to the motion equations for linear motion.Īngular velocity and angular acceleration A rigid body is an object with a mass that holds a rigid shape, such as a phonograph turntable, in contrast to the sun, which is a ball of gas. Rotational Motion of a Rigid Body Rotational motion is more complicated than linear motion, and only the motion of rigid bodies will be considered here.
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