A ball is tied to a string and whirled in a horizontal circle at a constant 4 revolutions per second. Categorize the following according to whether they are 0 in value, constant in value, or change:
v: linear speed (scalar)
v: linear velocity (vector)
ar: radial (centripetal) acceleration (scalar)
ar: radial (centripetal) acceleration (vector)
ω: angular speed (scalar)
ω: angular velocity (vector)
α: angular acceleration (scalar)
α: angular acceleration (vector)
I'm just pretty confused
Based on the given information, the categorization would be as follows:
v: linear speed (scalar) - constant in value
v: linear velocity (vector) - constant in value
ar: radial (centripetal) acceleration (scalar) - constant in value
ar: radial (centripetal) acceleration (vector) - constant in value
ω: angular speed (scalar) - constant in value
ω: angular velocity (vector) - constant in value
α: angular acceleration (scalar) - 0 in value
α: angular acceleration (vector) - 0 in value
To categorize the given quantities, let's understand their definitions and characteristics:
1. v: linear speed (scalar) - Linear speed refers to the magnitude or size of the velocity vector. It represents how fast an object is moving in a straight line. In this scenario, since the ball is whirling in a horizontal circle, its linear speed will not be constant. It changes because its magnitude and direction vary as it moves around the circle. Hence, linear speed is changing in value.
2. v: linear velocity (vector) - Linear velocity is the rate of change of displacement with respect to time and has magnitude and direction. In this case, the linear velocity of the ball will also change because its direction constantly changes as it moves around the circle. So, linear velocity is changing in value and direction.
3. ar: radial (centripetal) acceleration (scalar) - Radial acceleration, also known as centripetal acceleration, is the acceleration directed toward the center of the circular path. For an object moving in a circle at a constant speed, the magnitude of its centripetal acceleration remains constant, but its direction always points toward the center of the circle. Hence, radial acceleration is constant in value but changes direction.
4. ar: radial (centripetal) acceleration (vector) - The vector form of radial acceleration has both magnitude and direction. Similar to the scalar version, the magnitude of the radial acceleration will remain constant since the ball is moving in a circle at a constant speed. Its direction, however, will continuously change as it moves around the circle. Therefore, radial acceleration is constant in magnitude but changes direction.
5. ω: angular speed (scalar) - Angular speed measures how quickly an object rotates about an axis. It is the rate at which the angle changes as the object moves in a circular path. In this case, the ball is whirling in a horizontal circle at a constant 4 revolutions per second. Since the number of revolutions per second is constant, the angular speed will also be a constant value.
6. ω: angular velocity (vector) - Angular velocity is a vector quantity that represents the rate of change of angular displacement with respect to time. It has both magnitude and direction. In this scenario, the ball is whirling in a horizontal circle at a constant speed. As a result, its angular velocity will be constant both in magnitude and direction because the ball is rotating at a consistent rate.
7. α: angular acceleration (scalar) - Angular acceleration measures how quickly the angular velocity of an object changes over time. Angular acceleration occurs when an object's angular velocity changes, either by speeding up or slowing down its rotation. In this situation, the ball is whirling at a constant speed, so its angular acceleration is zero. There is no change in the angular velocity.
8. α: angular acceleration (vector) - Angular acceleration in vector form would have magnitude and direction. However, since the angular acceleration is zero in this scenario (due to the constant angular speed), both the magnitude and direction will be zero. Therefore, the angular acceleration is zero in value and direction.
In summary:
- Changing in value: v (linear speed), v (linear velocity)
- Changing in value and direction: v (linear velocity), ar (radial acceleration), ω (angular velocity)
- Constant in value: ar (radial acceleration - scalar)
- Constant in value and direction: ω (angular speed)
- Zero in value: α (angular acceleration - scalar), α (angular acceleration - vector)