A phonograph record initially moving at 33 1/3 rpm accelerates uniformly to 45 rpm in 2.5 s. What is the angular distance traveled by the record during this time?
Vo = 33.3333rev/min * 6.28rad/rev 1mim/60s = 3.49rad/s.
Vf = (45/33.3333) * 3.49rad/s=4.71rad/s
a = (Vf - Vo) / t,
a = (4.71 - 3.49) / 2.5 = 0.488.
d = Vo*t + 0.5a*t^2,
d = 3.49*2.5 +.244*(2.5)^2 = 10.25rad.
To find the angular distance traveled by the record, we can use the formula:
θ = ω1t + (1/2)αt²
Where:
θ is the angular distance
ω1 is the initial angular velocity
t is the time
α is the angular acceleration
Given:
ω1 = 33 1/3 rpm = 33.33 rpm
Final angular velocity, ω2 = 45 rpm
t = 2.5 s
To find α, we can use the formula:
α = (ω2 - ω1) / t
α = (45 - 33.33) rpm / 2.5 s
Now, we convert α to radians per second squared (rad/s²) since rpm is not in SI units.
1 revolution = 2π radians
1 minute = 60 seconds
α = ((45 - 33.33) revolutions / 2.5 seconds) * (2π radians / 1 revolution) * (1 minute / 60 seconds)
α = 0.1077 rad/s²
Now, we can substitute the values into the first formula to find θ.
θ = (33.33 rpm * 2.5 s) + (1/2)(0.1077 rad/s²)(2.5 s)²
Simplifying:
θ = 83.325 radians
Therefore, the angular distance traveled by the record during this time is approximately 83.325 radians.
To find the angular distance traveled by the record, we need to first calculate the angular acceleration of the record.
Given:
Initial speed (ω1) = 33 1/3 rpm = 33 1/3 * 2π rad/min
Final speed (ω2) = 45 rpm = 45 * 2π rad/min
Time (t) = 2.5 s
We know that angular acceleration (α) can be calculated using the formula:
α = (ω2 - ω1) / t
Substituting the given values:
α = (45 * 2π - 33 1/3 * 2π) / 2.5
Now we can simplify and calculate α:
α = (90π - 66.67π) / 2.5
α = 23.33π / 2.5
α = 9.3328 rad/s^2
Next, we can use the formula for angular displacement (θ) when uniformly accelerated motion is involved:
θ = ω1 * t + 0.5 * α * t^2
Substituting the given values:
θ = (33 1/3 * 2π) * 2.5 + 0.5 * 9.3328 * (2.5)^2
Now, we can simplify and calculate θ:
θ = 83.33π * 2.5 + 0.5 * 9.3328 * 6.25
θ ≈ 207.3125π
Hence, the angular distance traveled by the record during this time is approximately 207.3125π radians.