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1-Deriving centripetal acceleration (shortened)

For no apparent reason, a poodle is running counter-clockwise at a constant speed of 5.70m/s in a circle with radius 2.7m . Let v⃗ 1 be the velocity vector at time t1, and let v⃗ 2 be the velocity vector at time t2. Consider Δv⃗ =v⃗ 2−v⃗ 1 and Δt=t2−t1. Recall that a⃗ av=Δv⃗ /Δt.

Hint: It may be helpful to assume that at time t1, the poodle is on the x-axis, i.e., that the velocity vector v¯1 points along the y-axis.

Part A

For Δt = 0.4s calculate the magnitude (to four significant figures) of the average acceleration a⃗ av.

Express your answer using four significant figures.

aav =

m/s2

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Part B

For Δt = 0.4s calculate the direction (relative to v⃗ 1) of the average acceleration a⃗ av.

α =

∘

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Part C

For Δt = 4×10−2s calculate the magnitude (to four significant figures) of the average acceleration a⃗ av.

Express your answer using four significant figures.

aav =

m/s2

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Part D

For Δt = 4×10−2s calculate the direction (relative to v⃗ 1) of the average acceleration a⃗ av.

α =

∘

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Part E

Compare your results to the general expression for the instantaneous acceleration a⃗ for uniform circular motion that is derived in the text.

Compare your results to the general expression for the instantaneous acceleration for uniform circular motion that is derived in the text.

the magnitude of a⃗ av is less than of a⃗ and a⃗ av tends to a⃗ as Δt is decreased.

or

the magnitude of a⃗ av is greater than of a⃗ and a⃗ av tends to a⃗ as Δt is decreased.

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2-Acceleration of a rock in wheel tread

A rock stuck in the tread of a 57.0-cm-diameter bicycle wheel has a tangential speed of 3.50m/s . When the brakes are applied, the rock's tangential deceleration is 1.40m/s2 .

Part A

What is the magnitude of the rock's angular velocity at t = 1.60s ?

ω =

rad/s

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Part B

What is the magnitudes of the rock's angular acceleration at t = 1.60s ?

α =

rad/s2

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Part C

At what time is the magnitude of the rock's acceleration equal to g? (Remember that you need to include both tangential and radial accelerations in computing the magnitude of the rock's acceleration.)

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