Question

# Assessment Identify the Variables! In rotational kinematics - the variables are: t = time, which is...

Assessment

Identify the Variables!

In rotational kinematics - the variables are:

• t = time, which is measured in s (for seconds)
• θ = angle = what angle did the object turn thru, usually measured radians
• ωO = initial angular velocity = the rotational speed of the object at the beginning of the problem, which is measured in rad/s
• ω = final angular velocity = the rotational speed of the object at the end of the problem, which is measured in rad/s
• α = angular acceleration = how fast the object is speeding up or slowing down in its rotation, which is measured in rad/s2

For each problem, you need to find each numbers and assign it to one of these five variables. Here's an example problem:

"a motor is initially at rest and it picks up speed at 1.2 rad/s2 for 12 s until it reaches an angular velocity of 14 rad/s, in which time it turns thru 173 revolutions"

then you need to identify the following variables:

• t = 12 s
• θ = 173 revolutions, convert to radians = 173 rev (2π rad / rev) = 1087 rad

You have two things that help you nail down which number goes with what variable:

• the units of the each number in the problem will always tell exactly what the variable is and help give it the proper name - they work just like the regular kinematics units
• the context of the problem helps make the identification of the numbers clear. The problems we do are real life, everyday, familiar occurrences - try to see it in your mind and even draw a quick sketch of the problem including the variables to help you understand what's happening in the problem.

Solve the Formula and the Problem!

A typical problem might read "a wheel is turning at 6 rad/s when it speeds up at 2 rad/s2 for 3 s. Find the angle it turns thru"

First: find out what we already know

• t = 3 s

Second: figure out what we want to find out

• θ = ?

Third: find a formula that has those four variables - ωO , α, t and θ

• θ = ωOt + 1/2 αt2

Fourth: if the problem needs to be algebraically solved for the missing variable, do it. Then plug in the numbers and solve. In this problem we don't need to rearrange the formula as θ is already by itself.θ =(6 rad/s) (3 s) + 1/2 (2 rad/s2 )(3 s)2

Question 1 (1 point)

Match the following numbers with the right variable name - remember, the units are your best friend!

A gear moving at 2 rad/s accelerates at 1.4 rad/s2 to a speed of 12 rad/s. Find the angle the gear moves through.

Question 1 options:

 1 θ = 2 ωo = 3 ω = 4 α = 5 t =

Question 2 (1 point)

Same problem: A gear moving at 2 rad/s accelerates at 1.4 rad/s2 to a speed of 12 rad/s. Find the angle it moves thru.

Looking at the variables named in the last problem, which formula would be the best to use?

Question 2 options:

 A) ω = ωO + αt
 B) ω = θ / t
 C) ω2 = ωO2 + 2αθ
 D) θ  = ωOt + 1/2 αt2

Question 3 (1 point)

Same problem: A gear moving at 2 rad/s accelerates at 1.4 rad/s2 to a speed of 12 rad/s. Find the angle the gear moves through.

Question 3 options:

Question 4 (1 point)

A wheel spins at 15 rad/s. How long does it take to turn thru 500 rad?

Question 4 options:

 A) 33 s
 B) 7500 s
 C) 515 s
 D) 4,500,923.248 s

Question 5 (1 point)

A drill is moving at 6 rad/s when it speeds up at 2 rad/s2 for 3 s. Find the angular velocity at the end of the problem

Question 5 options:

Question 6 (1 point)

A motor is accelerating at 0.6 rad/s2 while it covers 20 rad. If it reaches a final angular velocity of 8 rad/s, find the initial angular velocity.

Question 6 options:

Question 7 (1 point)

An rotating disk starts from rest and accelerates at 1 rad/s2 for 17 s until it takes off. Determine the angle.

Question 8 (1 point)

An merry-go-round starts from rest and has an angular acceleration of 2 rad/s2 and covers 390 rad. Determine the time it takes.

Question 9 (1 point)

Note, the formula will still have t and t2 so it's a quadratic equation, and we can use the quadratic formula:

Steps:

• Simplify as much as possible and arrange in descending order - that is, get everything on one side of the equation with a zero on the other side and rearrange the terms so that t2 comes first, t comes second and the constant term is last
• the quad formula has a, b, and c which are the coefficients of the t2 term, the t term and the constant term including the positive or negative signs
• example:
• If 4 = 2 t + 1/2 (8) t2
• Simplify 1/2 (8) = 4, subtract 4 from both sides and rearrange the terms to get 0 = 4t2 + 2t - 4
• now a = 4, b = 2, c = -4
• plug these into the quadratic formula and get the two solutions, one when we use the + and the other when we use the - in the formula, just before the square root. Usually we want the positive solution
• (-2+√(22-4(4)(-4))).(2*4) = 0.78
• (-2-√(22-4(4)(-4))).(2*4) = -1.3

Question 10 (1 point)

Sometimes you have to do the problem in two steps:

A car moving at 22 rad/s comes to a stop in 12 s. Find the angle it moves thru.

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