×
Back to all chapters

Simple Harmonic Motion

A humbling fraction of physics boils down to direct application of simple harmonic motion, the description of oscillating objects. Learn the basis for springs, strings, and quantum fields.

Simple Harmonic Motion: Level 4-5 Challenges

Consider the earth as a uniform sphere of mass $$M$$ And Radius $$R$$. Imagine a straight smooth tunnel made through the earth which connects any two points on its surface (The two points are not diametrically opposite). Determine the time that a particle would take to go from one end to the other through the tunnel. If Your answer is $$T s$$, then find the value of $$\frac{T}{100}$$ to the nearest integer.

Details And Assumptions:

• Consider only gravitation force due to earth be acting on the particle during its motion along the tunnel.

• Take acceleration due to gravity on surface of Earth = $$10ms^{-2}$$

• Radius of Earth = $$6400 km$$.

This is an entry for the problem writing party.

A block of mass $$m$$ is gently attached to the spring and released at time $$t=0$$, when the spring has its free length. During subsequent motion of the block, the displacement of the block $$x$$ with respect to time $$t$$ is considered. Find $\displaystyle\int_0^3 x ~\mathrm{d}t$

Details and Assumptions

• Neglect air resistance.
• $$k$$ is the spring constant.
• $$m = 10\text{ kg},k = 34 \text{ Nm}^{-1}, g = 9.8 \text{ ms}^{-2}$$

This Question is not Original

A mass is subjected to a force $F = (at - bx) \text{ along the } x\text{-axis}$ Initially the mass lies at the origin at rest. In the definition of force (given) $$x$$ refers to the $$x$$-coordinate of the mass and t refers to the time elapsed.

Find the x - coordinate of the mass after a time of 4 seconds.

Assumptions
1) All the values are in SI units.
2) Take the mass = 1 kg, a = 1 N/s, b = 1 N/m

If the time period of SHM of rectangular block can be represented as $T=2\pi \sqrt{\frac{m(\sqrt a +b)}{ck}}$

Find the value of $$a+b+c$$.

Details and Assumptions:

• Treat the pulley as a disc.

• The block and pulley are oscillating in the same phase with the same frequency.

• Pulley has sufficient friction for Pure Rolling and mass=m & strings are light and inextensible.

• Gravity is present.

• Zig-zag lines in fig represents spring of spring constant $$k$$ and $$2k$$ as shown.

• a is a square-free integer and the fraction is in simplest form.

Try its different variants - medium and easy

A smooth wedge of mass $$m$$ and angle of inclination $$\theta = \SI{60}{\degree}$$ is attached to two springs of spring constant $$k_\textrm{L}$$ on the left, and $$k_\textrm{R} = 3k_\textrm{L}$$ on the right. The wedge rests on a smooth frictionless plane. Find the period of oscillation of the wedge in seconds.

Details and Assumptions:

• The springs are perpendicular to the respective sides they are facing.
• The spring on the left is constrained to compress and extend along its length. It is attached to the wedge by a frictionless roller that can move along the hypotenuse.
• $$m=\SI{3}{\kilo\gram}$$
• $$k_\textrm{L}= \frac18 \si[per-mode=symbol]{\newton\per\meter}$$.
×