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Hello everyone,

I got two different values for final angular speed when I tried to use the third and fourth kinematic equations. I filled the template here with the problem with my attempts.

1. The problem statement, all variables and given/known data

A bicycle wheel of radius r = 1.5 m starts from rest and rolls 100 m without slipping in 30 s. Assuming that the angular acceleration of the wheel given above was constant, calculate: a) The angular acceleration, b) the final angular velocity c) the tangential velocity and tangential acceleration of a point on the rim after one revolution.

Given

[tex]\Delta\theta = 100/1.5 = 66.7[/tex] rad

[tex]\omega_{i} = 0 [/tex]

[tex]\Delta t = 30 s[/tex]

[tex]\alpha[/tex] is constant

Unknowns

[tex] \alpha = ? [/tex]

[tex] \omega = ? [/tex]

[tex] v_{t} = ? [/tex] and [tex] a_{t} = ? [/tex] when [tex] \Delta\theta = 2\pi[/tex]

2. Relevant equations

[tex]\Delta\theta = \frac{1}{2} (\omega_{f} + \omega_{i}) \Delta t[/tex]

[tex] \omega_{f}^{2} = \omega_{i}^2 + 2 \alpha \Delta \theta [/tex]

[tex] \omega_{f} = \omega_{i} + \alpha \Delta t [/tex]

3. The attempt at a solution

Part b:

[tex]\Delta\theta = \frac{1}{2} (\omega_{f} + \omega_{i}) \Delta t[/tex]

[tex] 66.7 = \frac{1}{2}(0 + \omega_{f})(30) [/tex]

[tex] \omega_{f} = 4.45 [/tex] rad/s

Part a:

[tex] \omega_{f} = \omega_{i} + \alpha \Delta t [/tex]

[tex] 4.45 = 0 + 30\alpha [/tex]

[tex] \alpha = 0.15[/tex] rad/s^2

Part c:

My strategy is to find the final angular velocity [tex]\omega_f[/tex] using the first relevant equation. Then finding the tangential velocity by multiplying [tex]\omega_f[/tex] with the radius.

[tex]\Delta\theta = \frac{1}{2} (\omega_{f} + \omega_{i}) \Delta t[/tex]

[tex] 2\pi = \frac{1}{2} (\omega_{f} + 0)(30) [/tex]

[tex] \omega_{f} = \frac{2\pi}{15} = 0.419 [/tex]

And then:

[tex] v_{t} = r\omega = (1.5)(0.419) = 0.6285 [/tex] m/s

[tex] a_{t} = r\alpha = (1.5)(0.15) = 0.255 [/tex] m/s^2

However, the solution key has a different answer. [tex]\omega_{f}[/tex] after one revolution equals 1.37 rad/s and [tex]v_{t}[/tex] = 2.06. Apparently, they used this equation:

[tex] \omega_{f}^{2} = \omega_{i}^2 + 2 \alpha \Delta \theta [/tex]

I'm a little confused. I double checked my calculations, and I am almost sure that I haven't made a calculation error. I appreciate it if someone could help me with this.

Thanks,

Jin

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# Homework Help: Different values with rotational kinematic equations

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