PHYS 232 QUESTION BANK 1

1. Write down the general form of the linear differential equation of second order. Distinguish
between homogenous and non-homogeneous linear differential equations. If our aim is to solve
this equation, then briefly explain what this means.
2. Then write down linear differential equation of second order with constant coefficients. What
is the difference between this equation and the one you have written above?
3. Now write down the homogeneous linear differential equation of second order with constant
coefficients.
4. Write down the auxiliary equation of a homogeneous linear second order differential equation
with constant coefficients. Write down the general form of the solution of the above linear
differential equation after solving for the roots of the characteristic equation. Determine the
coefficients of the general solution by using the initial conditions
5. Write down the general solution, due to the principal of superposition, to the characteristic
equation for the homogeneous linear differential equation of second order with constant
coefficients.
6. Fill in the following table
TYPE OF HARMONIC
ROOTS m1 AND m2 ARE SOLUTION
MOTION
Real and distinct
Complementary and complex
Equal (repeated)

7. List the three mechanisms responsible for energy loss of a harmonic oscillator.
8. Which mechanism is used in our analysis of free damped simple harmonic motion and why?
9. Write down the differential equation for a system with a mass m, stiffness k, and a damping c.
10. Write down the same differential equation again in terms of damping ratio  and the un damped

angular frequency n .
11. If the damping is responsible negligible or zero what does the above differential equation
reduce to and what type of motion does it represent?

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12. What is the equation, which is obtained by substituting the trial solution into the damped
differential equation called?
a. Write down the roots to this equation. Which parameter in this equation determines
the type of roots?
b. List the three types of solution to the damped differential equation and how they
are categorised by the damping factor  .
c. List the three types of motion possible by this system and write their general
displacement functions.

13. Write down the expression for angular frequency  d ,. Is  d , greater or less than  n ?
14. Write down an expression for Td :

d. in terms of  d , and

e. In terms of n .
15. What is the difference between the amplitude in free shm and damped shm?
16. The displacement of a particle at t = 0.25 s is given by the expression 𝑥 =
(4.0 𝑚) 𝑐𝑜𝑠 (3.0 𝜋 𝑡 + 𝜋), where x is in meters and t is in seconds. Determine
(a) the frequency and period of the motion,
(b) the amplitude of the motion,
(c) the phase constant, and
(d) the displacement of the particle at t = 0.25 s
17. Distinguish between displacement and amplitude in damped simple harmonic motion.
18. What happens to the period of a simple pendulum if the pendulum's length is doubled? What
happens to the period if the mass of the bob is doubled?
19. A block-spring system undergoes simple harmonic motion with an amplitude A. Does the total
energy change if the mass is doubled but the amplitude is not changed? Do the kinetic and
potential energies depend on the mass?
20. A simple pendulum is 5.0 m long.
(a) What is the period of simple harmonic motion for this pendulum if it is located in an
elevator accelerating upward at 5.0 m/s2?
(b) What is the answer to part (a) if the elevator is accelerating down ward at 5.0 m/s 2 ?

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 (c) What is the period of simple harmonic motion for this pendulum if it is placed in a truck
that is accelerating horizontally at 5.0 m/s2 ?
21. A torsional pendulum is formed by attaching a wire to the center of a meter stick with mass
2.0 kg. If the resulting period is 3.0 minutes, what is the torsion constant for the wire?
22. A spring stretches by 3.9 cm when a 10-g mass is hung from it. If a 25-g mass attached to this
spring oscillates in simple harmonic motion, calculate the period of motion

23. What is the difference between free simple harmonic motion and damped simple harmonic
motion in terms of amplitude?
24. How much time is there between two consecutive amplitudes of the same sign in damped shm?
25. Define the logarithmic decrement
26. Define the relaxation time.
27. A body of mass 5.5 kg is hung on a spring of stiffness 1000 Nm-1. It is pulled down 50 mm
below the position of static equilibrium and released so that it executes vertical vibrations.
There is a viscous damping force acting on the body , of 40 N when the velocity is 1 ms -1.
a. Determine the differential equation of motion and obtain the expression for the
displacement of the body as a function of time.
b. Calculate the distance the body moves from the instant of release until it is
momentarily at rest at the highest pint of its travel and the time that has elapsed
when it reaches that position.
c. Calculate the time that elapses for the body to pass through the equilibrium position
for the first time after release.
28. A vibrating system consists of a 5kg mass, a spring with constant k=3.5Nmm-1, and a dashpot
with damping constant c=100 Nsm-1. Determine:
a. The damping factor (ratio)

b. The damped natural frequency d

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 The signal from a displacement transducer detecting displacements of a freely vibrating body is
in the form of a decaying sinusoid, as shown in the diagram below. Calculate the stiffness of the
elastic support for the vibrating mass of 4 kg, the damping constant and the amplitude h of the
displacement.

y/mm

0.3

3.2 t/s
3.8

-0.15

29. An industrial buffer unit comprises a piston of mass 5 kg, a spring of stiffness 2 kNm⁻¹, and a
damper that provides a linear damping force of 240 N at 1 m/s⁻¹. The unit forms a mass-spring-
damper system with a single degree of freedom as shown below:

The spring is initially compressed by 20 mm.

i. Obtain the differential equation for the motion of the piston as it moves to the
left in free motion and hence determine the damping ratio of the system.

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 ii. If a sudden impulse of 10 Ns is applied to the piston, determine the distance
moved by the piston before it momentarily comes to rest.

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