aspinning to has dimeter of1cm apoint on the out rim of the top moves through an angle of 8byradians each second. a,what is the angular velocity of the point? b,what is distance moved by the point in 5second. c,what is the velocity of the point? d,what is that acceleration of the point?
To answer these questions, we need to understand the concepts of angular velocity, distance moved, velocity, and acceleration in circular motion. Let's break down each question one by one and explain how to calculate the answers.
a) Angular velocity (ω) is the rate at which an object rotates around a fixed point. It is measured in radians per second (rad/s). In this case, we are given that the point on the outer rim of the spinning top moves through an angle of 8π radians each second.
To find the angular velocity (ω), we divide the angle moved by the time taken:
ω = angle moved / time taken
= 8π radians / 1 second
= 8π rad/s
So, the angular velocity of the point is 8π rad/s.
b) The distance moved by the point in 5 seconds can be calculated using the formula:
distance = angular velocity × time
In this case, the angular velocity is 8π rad/s, and the time is 5 seconds. Plugging these values into the formula:
distance = 8π rad/s × 5 seconds
= 40π cm
Therefore, the point moves a distance of 40π cm in 5 seconds.
c) Velocity (v) is the rate at which an object changes its position. It can be calculated by multiplying the radius (r) of the spinning object by the angular velocity (ω):
velocity = radius × angular velocity
In this case, the spinning top has a diameter of 1 cm, which means the radius (r) is half of that, i.e., 0.5 cm. The angular velocity (ω) is 8π rad/s. Plugging these values into the formula:
velocity = 0.5 cm × 8π rad/s
= 4π cm/s
Hence, the velocity of the point is 4π cm/s.
d) Acceleration (a) is the rate at which velocity changes over time. In circular motion, the acceleration is directed towards the center of the circle and can be calculated using the formula:
acceleration = radius × (angular velocity)^2
In this case, the radius (r) is 0.5 cm, and the angular velocity (ω) is 8π rad/s. Plugging these values into the formula:
acceleration = 0.5 cm × (8π rad/s)^2
= 32π^2 cm/s^2
Therefore, the acceleration of the point is 32π^2 cm/s^2.