(a) For the rotating mass in the lab setup by what ratio will the velocity change if the period is quadrupled?
(b) For the rotating mass in the lab setup by what ratio will the velocity change if the period is decreased by a factor of four?
(d) For the rotating mass in the lab setup by what ratio will the velocity change if the radius is decreased by a factor of four?
To answer these questions, we need to understand the relationship between velocity, period, and radius in the context of rotating mass in a lab setup. Let's break down each question step by step.
(a) For the rotating mass in the lab setup, to determine the ratio by which the velocity changes if the period is quadrupled, we can use the formula for velocity in circular motion:
Velocity = (2π * Radius) / Period
Let's assume the initial velocity is V1 and the initial period is T1. We can represent the initial formula as:
V1 = (2π * Radius) / T1
If the period is quadrupled (T1 -> 4T1), the new velocity (V2) can be represented as:
V2 = (2π * Radius) / (4T1)
Now, to find the ratio of the new velocity to the initial velocity, we divide V2 by V1:
Ratio = V2 / V1
= ((2π * Radius) / (4T1)) / ((2π * Radius) / T1)
= (T1 / 4T1)
= 1/4
Therefore, if the period is quadrupled, the velocity will decrease by a ratio of 1/4.
(b) For the rotating mass in the lab setup, to determine the ratio by which the velocity changes if the period is decreased by a factor of four, we will use the same formula as in part (a):
Velocity = (2π * Radius) / Period
Let's again assume the initial velocity is V1 and the initial period is T1. We can represent the initial formula as:
V1 = (2π * Radius) / T1
If the period is decreased by a factor of four (T1 -> T1/4), the new velocity (V2) can be represented as:
V2 = (2π * Radius) / (T1/4)
= 8(2π * Radius) / T1
Now, to find the ratio of the new velocity to the initial velocity, we divide V2 by V1:
Ratio = V2 / V1
= (8(2π * Radius) / T1) / ((2π * Radius) / T1)
= 4
Therefore, if the period is decreased by a factor of four, the velocity will change by a ratio of 4.
(c) For the rotating mass in the lab setup, to determine the ratio by which the velocity changes if the radius is decreased by a factor of four, we will again use the same formula as in parts (a) and (b):
Velocity = (2π * Radius) / Period
Let's once more assume the initial velocity is V1 and the initial radius is R1. We can represent the initial formula as:
V1 = (2π * R1) / Period
If the radius is decreased by a factor of four (R1 -> R1/4), the new velocity (V2) can be represented as:
V2 = (2π * (R1/4)) / Period
= π * R1 / Period
Now, to find the ratio of the new velocity to the initial velocity, we divide V2 by V1:
Ratio = V2 / V1
= (π * R1 / Period) / ((2π * R1) / Period)
= 1/2
Therefore, if the radius is decreased by a factor of four, the velocity will change by a ratio of 1/2.
(a) To find the ratio by which the velocity changes when the period is quadrupled, we can use the equation for the relationship between angular velocity (ω) and period (T):
ω = 2π / T
If the period is quadrupled, it becomes 4 times its original value. Let's call the original period T₀, and the new period T₁ = 4T₀.
Using the equation, we can express the initial angular velocity and final angular velocity as:
ω₀ = 2π / T₀
ω₁ = 2π / T₁ = 2π / (4T₀) = (1/2) * (2π / T₀) = (1/2) * ω₀
The ratio of the velocities can be determined by comparing ω₁ to ω₀:
ω₁ / ω₀ = ((1/2) * ω₀) / ω₀ = 1/2
Therefore, the velocity will change by a ratio of 1/2 when the period is quadrupled.
(b) To find the ratio by which the velocity changes when the period is decreased by a factor of four, we can use a similar approach to part (a).
If the period is decreased by a factor of four, it becomes 1/4 of its original value. Again, let T₀ be the original period and T₁ be the new period, where T₁ = (1/4)T₀.
Using the equation, we can express the initial angular velocity and final angular velocity as:
ω₀ = 2π / T₀
ω₁ = 2π / T₁ = 2π / ((1/4)T₀) = (4/1) * (2π / T₀) = 4ω₀
The ratio of the velocities can be determined by comparing ω₁ to ω₀:
ω₁ / ω₀ = (4ω₀) / ω₀ = 4
Therefore, the velocity will change by a ratio of 4 when the period is decreased by a factor of four.
(c) To find the ratio by which the velocity changes when the radius is decreased by a factor of four, we can use the equation for the relationship between linear velocity (v) and radius (r) in circular motion:
v = ωr
where v is the linear velocity, ω is the angular velocity, and r is the radius.
Let's assume the initial radius is r₀. If the radius is decreased by a factor of four, the new radius will be r₁ = (1/4)r₀.
The linear velocities can be expressed as:
v₀ = ωr₀
v₁ = ωr₁ = ω((1/4)r₀) = (1/4)(ωr₀) = (1/4)v₀
The ratio of the velocities can be determined by comparing v₁ to v₀:
v₁ / v₀ = ((1/4)v₀) / v₀ = 1/4
Therefore, the velocity will change by a ratio of 1/4 when the radius is decreased by a factor of four.