In the figure shown, the spring are connected to the rod at one end and at the midpoint. The rod is hinged at its lower end. (i) Find the minimum value of k for rotational SHM of the rod (Mass m , length L ) (ii) If k = mg L − 1 then find the angular frequency of oscillations of the rod.

(i) The rod performs SHM if the angular displacement of the rod is small (assumed) Restoring torque > Disturbing torque also for a small angular displacement we have θ = x 1 l / 2 , x 1 = θ l 2 , f 1 = K θ l 2 θ = x 2 l , x 2 = θ l , f 2 = K θ l τ = L 2 × F 1 × L × F 2 ≥ m g sin θ L 2 ⇒ L 2 K θ L 2 + L K θ L ≥ m g θ L 2 ⇒ 5 4 K θ L 2 ≥ m g θ L 2 ⇒ K ≥ 2 m g 5 L (ii) If the rod is displacement through an angle θ , then − k L θ L − k L 2 θ L 2 + M g L 2 θ = M L 2 3 α k = M g L ⇒ − 9 k θ 4 M = α ⇒ ω = 3 2 k m

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A massless road rigidly fixed at $O$ . A string carrying a mass $m$ at one end is attached to point $A$ on the rod so that $O A = a$ . At another point $B (O B = b)$ pf the rod, a horizontal spring of force constant $k$ is attached as shown. Find the period of small vertical oscillations of mass $m$ around its equilibrium position

Important Questions on Simple Harmonic Motion

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Beta Question Bank for Engineering: Physics>Chapter 19 - Simple Harmonic Motion>Exercise-4>Q 14

A life operator hung an exact pendulum clock on the lift wall in a lift in a building to know end of the working day. The lift moves with an upward & downward accelerations during the same time (according to a stationary clock), the magnitudes of the acceleration remaining unchanged. Will the operator work for more or less than required time.

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Beta Question Bank for Engineering: Physics>Chapter 19 - Simple Harmonic Motion>Exercise-4>Q 15

In the figure shown, the spring are connected to the rod at one end and at the midpoint. The rod is hinged at its lower end.

Question Image

(i) Find the minimum value of $k$ for rotational $SHM$ of the rod (Mass $m$ , length $L$ )

(ii) If $k = mg L^{- 1}$ then find the angular frequency of oscillations of the rod.

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Beta Question Bank for Engineering: Physics>Chapter 19 - Simple Harmonic Motion>Exercise-4>Q 16

Two identical balls

A

and

B

each of mass

0.1 kg

are attached to two identical massless springs. The spring mass system is constrained to move inside a rigid smooth pipe in the form of a circle as in fig. The pipe is fixed in a horizontal plane. The centres of the ball can move in a circle of radius

0.06 m

. Each spring has a natural length

0.06 πm

and force constant

0 .1 N m^{- 1}

. Initially both the balls are displaced by an angle of

θ = π / 6

radian with respect to diameter

P Q

of the circle and released from rest

Question Image

(i) calculate the frequency of oscillation of the ball $B$ .

(ii) what is the total energy of the system

(iii) find the speed of the ball $A$ when $A$ and $B$ are at the two ends of the diameter $P Q$ .

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Beta Question Bank for Engineering: Physics>Chapter 19 - Simple Harmonic Motion>Exercise-4>Q 17

A rod of mass

M

and length

L

is hinged at its one end and carries a block of mass

m

at its other end.

A

spring of force constant

k_{1}

is installed at distance a from the hinge and another of force constant

k_{2}

at a distance

b

as shown in the figure. If the whole arrangement rests on a smooth horizontal table top. find the frequency of vibration.

Question Image

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Beta Question Bank for Engineering: Physics>Chapter 19 - Simple Harmonic Motion>Exercise-4>Q 18

The following equation represent transverse wave;

$z_{1} = A \cos (k x - ω t), z_{2} = A \cos (k x + ω t), z_{3} = A \cos (k y - ω t)$

Identify the combination (s) of the waves which will produce. (i) standing wave (s) (ii) a wave travelling in the direction making an angle of $45 °$ with the positive $x$ and positive $y$ -axis. In each case, find the position at which the resultant intensity is always zero.

Important Questions on Simple Harmonic Motion

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