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Chapter 5: Problem 51

A car that is initially at rest moves along a circular path with a constant tangential acceleration component of \(2.00 \mathrm{m} / \mathrm{s}^{2} .\) The circular path has a radius of \(50.0 \mathrm{m} .\) The initial position of the car is at the far west location on the circle and the initial velocity is to the north. (a) After the car has traveled \(\frac{1}{4}\) of the circumference, what is the speed of the car? (b) At this point, what is the radial acceleration component of the car? (c) At this same point, what is the total acceleration of the car?

Short Answer

Expert verified
Question: A car starts from rest and accelerates tangentially at 2 m/s² on a circular path with a radius of 50 meters. Determine (a) the speed of the car, (b) the radial acceleration component, and (c) the total acceleration of the car after it has traveled 1/4 of the circumference. Answer: (a) The speed of the car after traveling 1/4 of the circumference is approximately 22.36 m/s. (b) The radial acceleration component of the car at this point is approximately 10.01 m/s². (c) At this same point, the total acceleration of the car is approximately 10.20 m/s².

Step by step solution

01

Determine the angle traveled (θ)

Since the car travels 1/4 of the circumference of the circle, we can determine the angle it has traveled in radians as follows: θ = (1/4) * 2π θ = π/2
02

Calculate the distance traveled (s)

We can calculate the distance traveled using its relationship with the radius and angle traveled: s = R * θ Here, R is the radius of the path and θ is the angle traveled in radians (which we already found in Step 1). s = 50 * (π/2) s ≈ 78.54 m
03

Calculate the final speed (vf)

We use the formula: \[v_f^2 = v_i^2 + 2as\] Where \(v_i\) is the initial speed, which is given as 0 m/s. a is the tangential acceleration, which is 2 m/s², s is the distance traveled, which is 78.54 m. \[v_f^2 = 0 + 2(2)(78.54)\] \[v_f \approx 22.36\:m/s\] (a) The speed of the car after traveling 1/4 of the circumference is approximately 22.36 m/s.
04

Calculate the radial acceleration (ar)

The radial (centripetal) acceleration can be calculated using this formula: \[a_r = \frac{v_f^2}{R}\] Where \(v_f\) is the final speed and R is the radius of the circular path. \[a_r = \frac{(22.36)^2}{50}\] \[a_r \approx 10.01\:m/s^2\] (b) The radial acceleration component of the car at this point is approximately 10.01 m/s².
05

Calculate the total acceleration (a_total)

The total acceleration can be determined by combining the radial acceleration (a_r) and the tangential acceleration (a_t) as given by the problem: \[a_{total} = \sqrt{a_r^2 + a_t^2}\] \[a_{total} = \sqrt{(10.01)^2 + (2)^2}\] \[a_{total} \approx 10.20\:m/s^2\] (c) At this same point, the total acceleration of the car is approximately 10.20 m/s².

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