Calculate the linear acceleration of a car, the 0.270-m radius tires of which have an angular acceleration of 15.0 rad/s2. Assume no slippage and give your answer in m/s2. m/s2 (b) How many revolutions do the tires make in 2.50 s if they start from rest? rev (c) What is their final angular velocity in rad/s? rad/s (d) What is the final velocity of the car in m/s? m/s

Answer: a) 4.05 m/s², b) 1.5625 rad, c) 37.5 rad/s, d) 10.125 m/s

Explanation:

a = linear acceleration = ?

Radius of car tires (r) = 0.27m

Angular acceleration (α) = 15.0 rad/s²

a)

The relationship between angular and linear acceleration is given by the formulae below

a = αr

a = 15.0 * 0.270

a = 4.05 m/s²

b)

Recall that

θ = ωot + αt²/2

Where θ = angular displacement and ωo = initial angular velocity = 0 (since the body starts from rest).

θ = αt²/2

θ = 15 * (2.5) ²/2

θ = 3.125/2 = 1.5625 rad.

But angular displacement = number of oscillations / time taken

1.5625 = number of oscillations / 2.5

Number of oscillations = 1.5625 * 2.5 = 3.91 rad.

c)

Recall that

ω = ωo + αt

But the body starts it motion from rest, hence ωo = 0

ω = 15 * 2.50

ω = 37.5 rad/s.

d)

Linear velocity is related to angular velocity via the formulae below

v = ωr

v = 37.5 * 0.270

v = 10.125 m/s

A) 4.05 m/s²

B) 7.46 revolutions

C) 37.5 rad/s

D) 10.125 m/s

Explanation:

A) Radius; r = 0.27m

Angular acceleration; α = 15 rad/s²

Now, the formula for linear acceleration is given by;

a = rα

Where r is radius and α is angular acceleration while a is linear acceleration.

Thus,

a = 0.27 x 15

a = 4.05 m/s²

B) First, Let's find the final velocity in rev/s using the equation of motion; V = u + αt

Where α is angular acceleration.

V = 0 + (15 x 2.5) = 37.5 rad/s

Now, we are calculating in revolutions, so let's convert the velocity to rev/s

1 rad/s = 1/2π rev/s

Thus, 37.5 rad/s = 37.5/2π revs/s = 5.968 rev/s

Now, average velocity = (v + u) / 2 = (5.968 + 0) / 2 = 5.968/2 = 2.984 revs/s

Thus, number of revolutions will be = average velocity x t =

2.984 rev/s x 2.5 s = 7.46 revolutions

C) we will get the final angular velocity from;

ω_f = ω_o + αt

Since it starts from rest, initial angular velocity ω_o = 0

Thus,

ω_f = 15 x 2.5 = 37.5 rad/s

D) The final velocity is given by;

v_f = ω_f•r = 37.5 x 0.27 = 10.125 m/s