Question: A wheel 2.00m in diameter lies in a vertical plane and rotates about its central axis with a constant angular acceleration of . The wheel starts at rest at t=0, and the radius vector of a certain point on the rim makes an angle of with the horizontal at this time. At, find (a) the angular speed of the wheel and, for point P, (b) the tangential speed, (c) the total acceleration, and (d) the angular position.
a. The angular speed of the wheel and, for point P is .
b. The tangential speed of the wheel is .
c. The total acceleration of the wheel is at
d. The angular position of the wheel is .
The formula that is used to find the angular speed is,
where is the initial position, is the angular acceleration, t is the time.
Hence, model the wheel as a rigid object under constant angular acceleration.
Use the formula to find the final angular speed at point P,
Substitute the given values , , .
Therefore, the angular speed of the wheel is .
Since the tangential speed can be expressed in terms of angular speed, the formula is
Substitute the given values, ,
Therefore, the tangential speed is .
The total acceleration is the sum of the tangential acceleration and radial acceleration that are perpendicular to each other.
Find the tangential acceleration of the wheel at that can be expressed in terms of angular acceleration.
Substitute the values, .
Determine the radial acceleration of the wheel that are expressed in terms of linear speed.
Substitute the values, .
Thus, the total acceleration can be found using the Pythagorean theorem, since both tangential and radial are perpendicular.
Hence, it makes an angle of radial acceleration vector of
Therefore, the total acceleration of the wheel is at .
For the rigid object under constant angular acceleration, the equation that is used to find angular position is
Substitute the values .
Thus, the angular position of the wheel is .
A ball having mass m is fastened at the end of a flagpole that is connected to the side of a tall building at point Pas shown in Figure. The length of the flagpole is, and it makes an angle with the x-axis. The ball becomes loose and starts to fall with acceleration . (a) Determine the angular momentum of the ball about point Pas a function of time. (b) For what physical reason does the angular momentum change? (c) What is the rate of change of the angular momentum of the ball about point P?
Question: The fishing pole in Figure P10.28 makes an angle of with the horizontal. What is the torque exerted by the fish about an axis perpendicular to the page and passing through the angler’s hand if the fish pulls on the fishing line with a force at an angle below the horizontal? The force is applied at a point from the angler’s hands.
Question: In a manufacturing process, a large, cylindrical roller is used to flatten material fed beneath it. The diameter of the roller is , and, while being driven into rotation around a fixed axis, its angular position is expressed as where is in radians and t is in seconds. (a) Find the maximum angular speed of the roller. (b) What is the maximum tangential speed of a point on the rim of the roller? (c) At what time t should the driving force be removed from the roller so that the roller does not reverse its direction of rotation? (d) Through how many rotations has the roller turned between t=0 and the time found in part (c)?
Use the definition of the vector product and the definitions of the unit vectors to prove Equations 11.7. You may assume the x axis points to the right, the y axis up, and the z axis horizontally toward you (not away from you). This choice is said to make the coordinate system a right-handed system.
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