An old-fashioned single-play vinyl record rotates on a turntable at 46.0 rpm. What are (a) the angular velocity in rad/s and (b) the period of the motion?
46rpm*2pirad/rev*1min/60sec=46*2*PI/60 rad/sec
Period is 1/46rpm=60sec/46rev*1rev= 60/46 sec
To calculate the angular velocity and period of the motion of the vinyl record, we need to convert the given rotational speed from revolutions per minute (rpm) to radians per second (rad/s) and seconds (s), respectively.
(a) The angular velocity (ω) is the rotational speed in radians per second. To calculate it, we can use the following formula:
ω = (2π * N) / T
where N is the number of revolutions per minute and T is the number of minutes in one second.
Given N = 46.0 rpm, we can substitute it into the formula:
ω = (2π * 46.0) / 60
Now, let's calculate:
ω = (2π * 46.0) / 60
ω ≈ 4.834 rad/s
Therefore, the angular velocity of the vinyl record is approximately 4.834 rad/s.
(b) The period (T) of the motion is the time it takes for one complete revolution. It can be calculated using the following formula:
T = 1 / (N/60)
where N is the number of revolutions per minute.
Given N = 46.0 rpm, we can substitute it into the formula:
T = 1 / (46.0/60)
Now, let's calculate:
T = 1 / (46.0/60)
T ≈ 1.304 s
Therefore, the period of the motion is approximately 1.304 seconds.
To find the angular velocity in rad/s, we need to convert the rotation rate from rpm (revolutions per minute) to rad/s (radians per second).
(a) Angular velocity (ω) is defined as the change in angle per unit time. We can convert rpm to rad/s using the following formula:
Angular velocity (rad/s) = (Angular velocity in rpm) × (2π rad/1 revolution) × (1 minute/60 seconds)
Substituting the given value:
Angular velocity (rad/s) = 46.0 rpm × (2π rad/1 revolution) × (1 minute/60 seconds)
Simplifying this expression will give us the angular velocity in rad/s.
(b) The period (T) of the motion, on the other hand, represents the time taken for one complete revolution. It is the reciprocal of the angular velocity.
Period (T) = 1 / Angular velocity (rad/s)
To calculate the period, we need to first calculate the angular velocity using the formula (a) and then take the reciprocal.
Now, let's perform the calculations:
(a) Angular velocity:
Angular velocity (rad/s) = 46.0 rpm × (2π rad/1 revolution) × (1 minute/60 seconds)
Angular velocity (rad/s) = (46.0 × 2π) / 60
(b) Period:
Period (T) = 1 / Angular velocity (rad/s)
Period (T) = 1 / ((46.0 × 2π) / 60)
By substituting the value of the angular velocity obtained in part (a) into the period formula, we can calculate the period of the motion.