Angular Kinematics

Question

Hello, I am struggling in my science class and I'm not sure how to do the unit we are on so I was wondering if someone could give me the answers and how to solve them so I can do better. Thank you! Here is the link to what I need help on: https://ibb.co/bLGKQJk

(this is a re-post because I ended up posting a link that expired)

Answer 1

For the FIRST TABLE you need two equations:

rev = 2π rad (A1)

s = Rθ (where θ is in rad) (B1)

As I will explain below,

equation (A1) allows you to calculate column 2 from column 1, and

equation (B2) allows you to calculate column 4 from columns 2 and 3.

These two equations can be rewritten to calculate other columns

rad = rev/2π (A2)

R = s/θ (B2)

θ = s/R (B3)

As an example, use (A1) to calculate θ(rad) from θ(rev) on line 1 of the table.

As it is with unit conversion in general, it is convenient to think of a unit as a number multiplying an amount:

17 rev = 17 × rev = 17 × 2π × rad = 34π × rad = 34π rad

Conversely you can use (A2) to calculate θ(rev) from θ(rad) on line 2:

2 rad = 2 × rad = 2 × rev/2π = 1/π rev

Using equations (A1) and (A2) you can similarly fill in the values of θ on the last two lines of the table.

On line 1, having calculated θ(rad) as 34π, we can calculate s using equation (B1).

s = Rθ = 0.5 × 34π = 17π

Similarly you can calculate s on line 2.

On line 3 use equation (B3)

θ = s/R = 12/3 = 4

That quantity is in radians, which fills in the second column of line 3.

And from that you calculate the first column using equation (A2)

For line 4 and 5 you calculate R using equation (B2).

For the SECOND TABLE I will deviate from the given notation to write "sec" for the

unit of time, not "s", in order to avoid any confusion with "s" used for distance in the

first table.

The equations for the second table can be derived from the equations of the first table:

RPM = rev/min = 2π×rad/60sec = π/30 rad/sec (C1)

v = s/t = Rθ/t = Rω (D1)

As in the first table, from (C1) and (D1) we derive

rad/sec = 30/π RPM (C2)

R = v/ω (D2)

ω = v/R (D3)

For example, you can calculate ω on line 1 using (C1):

40 RMP = 40 × RMP = 40 × π/30 × rad/sec = 4π/3 rad/sec

All the other entries in the second table can be obtained just like in the first table.