Question

A 4-centimeter rod is attached at one end to a point a rotating counterclockwise on a wheel of radius 2 cm. the other end b is free to move back and forth along a horizontal bar that goes through the center of the wheel. at time t=0 the rod is situated as in the diagram at the left below. the wheel rotates counterclockwise at 3.5 revolutions per second. thus, when t=1/21 sec, the rod is situated as in the diagram at the right below. note that the wheel has its center at the origin. what is the coordinate of a?

Answer 1

To find the coordinates of point A on the rotating wheel, we would use the radius and the angular velocity, calculating the angle covered to apply trigonometric functions to get the x and y coordinates.

Step-by-step explanation:

To address the student's question, we first calculate the angular velocity of the wheel. At 3.5 revolutions per second, the wheel rotates at an angular speed of ω = 2π × 3.5 rad/s. Since the wheel's radius is 2 cm, we can calculate the point A's position at any given time using the rotational kinematic equations. However, the student's description lacks concrete time and diagram to determine the exact position of A at time t. Assuming we had such data, we could use trigonometric functions. For example, if A is originally at the rightmost point at time t=0, at t=1/21 sec, the wheel has undergone an angle θ = ω × t radians. Using the angle, we calculate the coordinates of point A as:

x = r × cos(θ)
y = r × sin(θ)

Substitute ω = 2π × 3.5 rad/s, r = 2 cm, and t = 1/21 sec into θ to find the coordinates at the specified time.

Answer 2

The coordinate of point A on the rotating wheel is determined by calculating its angular displacement using the angular velocity and then applying trigonometric functions to find its position in Cartesian coordinates.

Step-by-step explanation:

The coordinate of point A on a wheel rotating counterclockwise can be found by understanding the rotation movement and using trigonometric functions. Assuming that point A moves in a circular path with the rotation of the wheel, you can determine its coordinate at any given time t by using the equations for circular motion.

To find the coordinate of point A at any time t, you need to know the angle \(\theta\) it rotates through, which can be found by the wheel's angular velocity \(\omega\). The angular displacement \(\theta\) is given by \(\omega t + \frac{1}{2} \alpha t^{2}\), where \(\alpha\) is the angular acceleration. Since the wheel rotates at a constant rate of 3.5 revolutions per second and there's no mention of angular acceleration, you can assume \(\alpha\) = 0. One revolution is \(2\pi\) radians, so the angular velocity in radians per second is \(\omega = 3.5 \times 2\pi\). Therefore, the angle of rotation after time t is \(\theta = \omega \cdot t\).

Considering the initial position of point A and using the sine and cosine functions, you can find the x and y coordinates of A. The x-coordinate is r \cdot cos(\theta) and the y-coordinate is r \cdot sin(\theta), where r is the radius of the wheel.

In conclusion, to find the coordinate of point A at any given time t, calculate the angle of rotation \(\theta\) and use trigonometric functions to get the x and y coordinates with respect to the radius of the wheel.