Question
Use the active figure depicting a rigid object that rotates around a fxed axis to complete the exercis.
Click 'start ' to begin the animation.
Instructions Use the Active Figure depicting a rigid object that rotates around a fixed axis to complete the exercise. Click 'start' to begin the animation. A rigid object rotates around a fixed axis through the origin, as shown in the Active Figure. The angle theta 's rate of change is omega = 1.5 rad/s. What is the speed of a point P in the object at a distance of r = 95 m from the axis of rotation? RECOGNIZE THE PRINCIPLES When a rigid body rotates through a given angle, all parts of the body rotate through the same angle at the same time. A point P located on the rigid body at a particular distance from the axis of rotation moves about the axis of rotation in a circle whose radius is the distance to the point. We are dealing with rotational We are dealing with rotational motion of a rigid body with a prescribed rate of rotation. To solve the problem, we need to know the relationship between the rotation rate of the object and the speed at which a particular point on the object moves as a result of the rotational motion. IDENTIFY THE RELATIONSHIPS AND SOLVE As the object rotates through a small angle delta theta in the short time delta t, the point P moves a distance delta s = r delta theta along the arc of its circular path. Therefore v = delta s/delta t = r delta theta /delta t. This becomes v = r omega where omega = d theta/dt. Therefore, the tangential speed of the point P is v = omega r = (1.5 rad/s)(95 m) = m/s. SUMMARY Use the simulation to examine the relationship between angular rate of rotation and linear speed. How does an increase or decrease in tangential speed translate into a change in angular speed? What if the distance from the axis is multiplied by a factor of 3.5 with the angular rate of rotation omega kept the same? The small distance delta s would correspondingly be multiplied by a factor of and therefore the tangential speed v would be multiplied by a factor of
Explanation / Answer
v = w*r = 1.5*95 = 142.5 units/sec
Angular rate of rotation is directly proportional to tangential speed.
It increases linearly with tangential speed
3.5, 3.5
