Mechanics Circular Motions

Mechanics Circular Motions

Mechanics Circular Motions are handled by resolving forces horizontally and vertically in conjunction with the tension of the string, normal reactions, circular motion, and the particle’s mass.

Worked Examples of Mechanics Circular Motions

A particle of mass \(m\) is attached to one end of a string of length \(R\). The other end of the string is fixed at height \(2h\) above the centre of a sphere of radius \(R\). The particle moves in a circle of radius \(r\) on the surface of the sphere and has constant angular velocity \( \omega > 0 \). The string makes an angle of \( \theta \) with the vertical.
Three forces act on the particle: the tension force \(F\) of the string, the normal reaction force \(N\) to the surface of the sphere, and the gravitational force \(mg\).

(a)    By resolving the forces horizontally and vertically, show that

$F \sin \theta-N \sin \theta = m \omega ^2 r \\
F \cos \theta + N \cos \theta = mg$

Resolve forces for Mechanics Circular Motions.
\( \begin{aligned} \displaystyle \require{AMSsymbols} \require{color}
F \sin \theta &= N \cos \theta + m \omega r^2 r &\color{red} \text{horizontally} \\
\therefore F \sin \theta-N \sin \theta &= m \omega ^2 r &\color{red} (1) \\
F \sin \theta + N \sin \theta &= mg &\color{red} \text{vertically} \\
\therefore F \cos \theta + N \cos \theta &= mg &\color{red} (2)
\end{aligned} \)

(b)    Show that \( \displaystyle N = \frac{1}{2} mg \sec \theta-\frac{1}{2} m \omega r \ \text{cosec} \ \theta \).

\( \begin{aligned} \displaystyle \require{AMSsymbols} \require{color}
\frac{F \sin \theta}{\sin \theta}-\frac{N \sin \theta}{\sin \theta} &= \frac{m \omega ^2 r}{\sin \theta} &\color{red} (1) \div \sin \theta \\
F-N &= m \omega ^2 r \ \text{cosec} \ \theta &\color{red} (3) \\
\frac{F \cos \theta}{\cos \theta} + \frac{N \cos \theta}{\cos \theta} &= \frac{mg}{\cos \theta} &\color{red} (2) \div \cos \theta \\
F + N &= mg \sec \theta &\color{red} (4) \\
2N &= mg \sec \theta-m \omega^2 r \ \text{cosec} \ \theta &\color{red} (4)-(3) \\
\therefore N &= \frac{1}{2} mg \sec \theta-\frac{1}{2} m \omega^2 r \ \text{cosec} \ \theta
\end{aligned} \)

(c)    Show that the particle remains in contact with the sphere if \( \displaystyle \omega \le \sqrt{\frac{g}{h}} \).

\( \newcommand\ddfrac[2]{\frac{\displaystyle #1}{\displaystyle #2}} \)
The particle stays in contact with the sphere \(N \ge 0\).
\( \begin{aligned} \displaystyle \require{AMSsymbols} \require{color}
\frac{1}{2} mg \sec \theta-\frac{1}{2} m \omega^2 r \ \text{cosec} \ \theta &\ge 0 \\
mg \sec \theta-m \omega^2 r \ \text{cosec} \ \theta &\ge 0 \\
– m \omega^2 r \ \text{cosec} \ \theta &\ge-mg \sec \theta \\
m \omega^2 r \ \text{cosec} \ \theta &\le mg \sec \theta \\
\omega^2 &\le \ddfrac{mg \sec \theta}{mr \ \text{cosec} \ \theta} \\
\omega^2 &\le \frac{mg \times \ddfrac{R}{h}}{mr \times \ddfrac{R}{r}} &\color{red} \sec\theta = \frac{R}{h}, \text{cosec} \ \theta = \frac{R}{r}\\
\omega^2 &\le \frac{g}{h} \\
\therefore \omega &\le \sqrt{\frac{g}{h}}
\end{aligned} \)

Unlock your full learning potential—download our expertly crafted slide files for free and transform your self-study sessions!

Discover more enlightening videos by visiting our YouTube channel!


Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume

Related Articles


Your email address will not be published. Required fields are marked *