Consider different points along one spoke of a wheel rotating with constant angular velocity. Which of the following is true regarding the centripetal acceleration at a particular instant of time?
A The magnitude of the centripetal acceleration is greater for points on the spoke closer to the hub than for points closer to the rim
B both the magnitude and the direction of the centripetal acceleration depend on the location of the point on the spoke.
C The magnitude of the centripetal acceleration is smaller for points on the spoke closer to the hub than for points closer to the rim but the direction of the acceleration is the same at all points on this spoke.
D The magnitude and direction of the centripetal acceleration is the same at all points on this spoke.
C. is the only correct statement
Well, if we're talking about a wheel rotating with constant angular velocity, it means that it's traveling at the same speed throughout. So, let's put on our thinking caps and analyze this!
Since the angular velocity is constant, that means the wheel is spinning at a steady rate. Now let's focus on the centripetal acceleration. Remember, centripetal acceleration is the acceleration towards the center of the circle.
Now, think about the points on the spoke. The points closer to the hub are closer to the center of the circle, while the points closer to the rim are farther away. This means that the points closer to the hub have a smaller radius compared to the points closer to the rim.
So, based on this, what do you think is true about the magnitude of the centripetal acceleration? Is it greater for points on the spoke closer to the hub or for points closer to the rim?
I'll give you a moment to think.
A The magnitude of the centripetal acceleration is greater for points on the spoke closer to the hub than for points closer to the rim.
To determine the true statement regarding the centripetal acceleration at a particular instant of time along a spoke of a rotating wheel, let's analyze the nature of centripetal acceleration and its relationship to the location on the spoke.
Centripetal acceleration is the acceleration experienced by an object moving in a circular path, directed towards the center of that path. It is always perpendicular to the velocity vector and points towards the center of the circle.
Now, for a rotating wheel, the angular velocity is constant, which means that all points on the spoke are moving with the same angular speed or rate. However, the linear speed of each point on the spoke varies according to its distance from the center (hub) and the radius of the wheel.
With this understanding, let's evaluate the given options:
A) The magnitude of the centripetal acceleration is greater for points on the spoke closer to the hub than for points closer to the rim.
This option is not generally true. The magnitude of the centripetal acceleration is determined by the square of the linear speed divided by the radius. Since the linear speed of a point on the spoke depends on its distance from the center, the centripetal acceleration will vary.
B) Both the magnitude and the direction of the centripetal acceleration depend on the location of the point on the spoke.
This option is generally true. As mentioned earlier, the magnitude of the centripetal acceleration depends on the linear speed of a point, which varies with the location (distance from the hub). The direction of the centripetal acceleration is always towards the center of the circle, regardless of the location.
C) The magnitude of the centripetal acceleration is smaller for points on the spoke closer to the hub than for points closer to the rim, but the direction of the acceleration is the same at all points on this spoke.
This option is also not generally true. As explained earlier, the magnitude of the centripetal acceleration depends on the linear speed, which varies with the location. Therefore, the magnitude of the centripetal acceleration will also vary.
D) The magnitude and direction of the centripetal acceleration are the same at all points on this spoke.
This option is generally true. Even though the magnitude of the centripetal acceleration varies with the location, the direction of the acceleration remains the same at all points on the spoke – directed towards the center.
Based on this analysis, option D is the correct answer.