Question 7.5: AT THE RACETRAK GOAL Apply the concepts of centripetal accel...
AT THE RACETRAK
GOAL Apply the concepts of centripetal acceleration and tangential velocity.
PROBLEM A race car accelerates uniformly from a speed of 40.0 \mathrm{~m} / \mathrm{s} to a speed of 60.0 \mathrm{~m} / \mathrm{s} in 5.00 \mathrm{~s} while traveling counterclockwise around a circular track of radius 4.00 \times 10^2 \mathrm{~m}. When the car reaches a speed of 50.0 \mathrm{~m} / \mathrm{s}, calculate (a) the magnitude of the car’s centripetal acceleration, (b) the angular velocity, (c) the magnitude of the tangential acceleration, and (d) the magnitude of the total acceleration.
STRATEGY Substitute values into the definitions of centripetal acceleration (Eq. 7.13),
a_c=\frac{v^2}{r} [7.13]
tangential velocity (Eq. 7.10),
v_t = rω [7.10]
and total acceleration (Eq. 7.18).
a = \sqrt{a_t^2 + a_c^2} [7.18]
Dividing the change in tangential velocity by the time yields the tangential acceleration.
(a) Calculate the magnitude of the centripetal acceleration when v=50.0 \mathrm{~m} / \mathrm{s}.
Substitute into Equation 7.13:
a_c=\frac{v^2}{r}=\frac{(50.0 \mathrm{~m} / \mathrm{s})^2}{4.00 \times 10^2 \mathrm{~m}}=6.25 \mathrm{~m} / \mathrm{s}^2
(b) Calculate the angular velocity.
Solve Equation 7.10 for \omega and substitute:
\omega=\frac{v}{r}=\frac{50.0 \mathrm{~m} / \mathrm{s}}{4.00 \times 10^2 \mathrm{~m}}=0.125 \mathrm{rad} / \mathrm{s}
(c) Calculate the magnitude of the tangential acceleration.
Divide the change in tangential velocity by the time:
a_t=\frac{v_f-v_i}{\Delta t}=\frac{60.0 \mathrm{~m} / \mathrm{s}-40.0 \mathrm{~m} / \mathrm{s}}{5.00 \mathrm{~s}}=4.00 \mathrm{~m} / \mathrm{s}^2
(d) Calculate the magnitude of the total acceleration.
Substitute into Equation 7.18:
\begin{aligned}&a=\sqrt{a_t^2+a_c^2}=\sqrt{\left(4.00 \mathrm{~m} / \mathrm{s}^2\right)^2+\left(6.25 \mathrm{~m} / \mathrm{s}^2\right)^2} \\&a=7.42 \mathrm{~m} / \mathrm{s}^2\end{aligned}
REMARKS We can also find the centripetal acceleration by substituting the derived value of \omega into Equation 7.17.
a_c = \frac{r^2 ω^2}{r} = r ω^2 [7.17]