A car of mass \(m\) kg moves round a curve of radius \(r\) m on a road which is banked at an angle \(\theta\) to the horizontal. When the speed of the car is \(u\) ms\(^{-1}\), the car experiences no sideways frictional force. Given that \(\tan \theta = \frac{u^2}{gr}\), show that the sideways frictional force on the car when its speed is \(\frac{u}{2}\) ms\(^{-1}\) has magnitude \(\frac{3}{4}mg \sin \theta\) N. [10 marks]

Speed $< u$, so friction $F$ acts up. $R \cos \theta + F \sin \theta = mg$ (1) | B1 M1 A1 |

$R \sin \theta - F \cos \theta = \frac{mu^2}{4r}$ (2) | M1 A1 A1 M1 |

(1) $\times \sin \theta$, (2) $\times \cos \theta$, subtract: | |

$F = mg \sin \theta - \frac{mu^2}{4r}\cos \theta = mg \sin \theta - \frac{u^2}{4}\tan \theta \cos \theta = \frac{3}{4}mg \sin \theta$ | A1 A1 A1 |

**Total: 10 marks**

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A car of mass $m$ kg moves round a curve of radius $r$ m on a road which is banked at an angle $\theta$ to the horizontal. When the speed of the car is $u$ ms$^{-1}$, the car experiences no sideways frictional force. Given that $\tan \theta = \frac{u^2}{gr}$, show that the sideways frictional force on the car when its speed is $\frac{u}{2}$ ms$^{-1}$ has magnitude $\frac{3}{4}mg \sin \theta$ N. [10 marks]

\hfill \mbox{\textit{Edexcel M3  Q3 [10]}}