Pearson Edexcel IAL Mathematics M3.4 Motion in a circle Question Bank

Practise circular-motion models for particles on cones, bowls and strings, resolving forces to link speed, radius and contact conditions.

Exam points

resolve horizontally and vertically for a particle moving in a horizontal circle
use radial equations with R, T or friction to prove inequalities for speed or ω^2
apply loss of contact or slack-string conditions through R=0 or T=0

Covered syllabus points

M3.4.1 - Angular speed M3.4.2 - motion M3.4.3 - Uniform motion of a particle M3.4.4 - Motion of a particle in a vertical circle

[Maximum number: 14]

Figure 5

A particle P of mass m is attached to one end of a light inextensible string of length $a \sqrt{3}$ . The other end of the string is attached to a fixed point A. The particle P is also attached to one end of a second light inextensible string of length a. The other end of this string is attached to a fixed point B, where B is vertically below A, with A B=a.

The particle P moves in a horizontal circle with centre O, where O is vertically below B.
The particle P moves with constant angular speed $\omega$ , with both strings taut, as shown in Figure 5.

(a)

Show that the upper string makes an angle of $30^{\circ}$ with the downward vertical and the lower string makes an angle of $60^{\circ}$ with the downward vertical.

[ 2 ]

(b)

Show that the tension in the upper string is $\frac{1}{2} m \sqrt{3}\left(2 g-a \omega^{2}\right)$ .

[ 8 ]

$T_A\cos30^\circ+T_B\cos60^\circ=mg$
M1 A1
$T_A\sin30^\circ+T_B\sin60^\circ=mr\omega^2$
M1 A1 A1
$r=a\sin60^\circ=\frac{a\sqrt3}{2}$
B1
Solve for $T_A$ .
dM1
$T_A=\frac12m\sqrt3(2g-a\omega^2)$
A1*

(c)

Show that $\frac{2 g}{3 a}<\omega^{2}<\frac{2 g}{a}$

[ 4 ]

Attempt to obtain one inequality on $\omega^2$ .
M1
Correct inequality.
A1
Attempt to obtain another inequality on $\omega^2$ and use both to obtain answer.
M1
$\frac{2g}{3a}<\omega^2<\frac{2g}{a}$
A1*
M1 correct use of either $T_A>0$ or their $T_B>0$ to obtain one inequality on $\omega^2$ .
A1 correct inequality.
M1 use both $T_A>0$ and their $T_B>0$ to form inequalities.
A1* given answer correctly obtained.

[Maximum number: 4]

Figure 5

A thin smooth hollow spherical shell has centre O and radius r. Part of the shell is removed to form a bowl with a plane circular rim. The bowl is fixed with the circular rim uppermost and horizontal. The point A is the lowest point of the bowl, as shown in Figure 5.

The point B is on the rim of the bowl, with OB at an angle $\theta$ to the upward vertical, where $\tan \theta=\frac{12}{5}$
A small ball is placed in the bowl at A. The ball is projected from A with horizontal speed u and moves in the vertical plane AOB. The ball stays in contact with the bowl until it reaches B.

At the instant when the ball reaches B, the speed of the ball is v.
By modelling the ball as a particle and ignoring air resistance,

show that $u^{2} \geqslant \frac{41}{13} g r$

\frac{mv^2}{r}\leq mg\cos\theta

v^2\leq gr\cos\theta=\frac{5gr}{13}

u^2-\frac{36gr}{13}\leq \frac{5gr}{13}

A1*

u^2\geq \frac{41gr}{13}

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Covered syllabus points

M3.4 - Motion in a circle question 1

Question (a)

Question (b)

Question (c)

M3.4 - Motion in a circle question 2