Advertisements
Advertisements
प्रश्न
A spring having with a spring constant 1200 N m–1 is mounted on a horizontal table as shown in Fig. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released.
Determine (i) the frequency of oscillations, (ii) maximum acceleration of the mass, and (iii) the maximum speed of the mass.
उत्तर १
Spring constant, k = 1200 N m–1
Mass, m = 3 kg
Displacement, A = 2.0 cm = 0.02 cm
(i) Frequency of oscillation v, is given by the relation:
`v = 1/T = 1/(2pi) sqrt(k/m)`
Where, T is the time period
`:. v = 1/(2xx3.14) sqrt(1200/3) = 3.18 "m/s"`
Hence, the frequency of oscillations is 3.18 cycles per second.
ii) Maximum acceleration (a) is given by the relation:
a = ω2 A
Where
ω = Angular frequency = `sqrt(k/m)`
A = Maximum displacement
`:. a = k/m A = (1200xx0.02)/(3) = 8 ms^(-2)`
Hence, the maximum acceleration of the mass is 8.0 m/s2
iii) Maximum velocity, vmax = Aω
`= A sqrt(k/m) = 0.02 xx sqrt(1200/3) = 0.4 "m/s"`
Hence, the maximum velocity of the mass is 0.4 m/s.
उत्तर २
K = 1200 `Mn^(-1)`; m = 3.0 kg, a= 2.0 cm = 0.02 m
i) Frequency, `v = 1/T = 1/(2pi) sqrt(k/m) = 1/(2xx3.14) sqrt(1200/3) = 3.2 s^(-1)`
ii) Acceleration, A = `omega^2` `" " y = k/m y`
Acceleration will be maximum when y is maximum i.e y = q
:. max acceleration,` A_"max" = (ka)/m =(1200xx0.02)/3 = 8 ms^(-2)`
iii) Max speed of the mass will be when it is passing throught mean position
`V_"max" = aomega = sqrt(k/m) = 0.02 xx sqrt(1200/3) = 0.4 ms^(-1)`
APPEARS IN
संबंधित प्रश्न
If the metal bob of a simple pendulum is replaced by a wooden bob of the same size, then its time period will.....................
- increase
- remain same
- decrease
- first increase and then decrease.
When the length of a simple pendulum is decreased by 20 cm, the period changes by 10%. Find the original length of the pendulum.
Answer the following questions:
A man with a wristwatch on his hand falls from the top of a tower. Does the watch give correct time during the free fall?
The cylindrical piece of the cork of density of base area A and height h floats in a liquid of density `rho_1`. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period
`T = 2pi sqrt((hrho)/(rho_1g)`
where ρ is the density of cork. (Ignore damping due to viscosity of the liquid).
Show that motion of bob of the pendulum with small amplitude is linear S.H.M. Hence obtain an expression for its period. What are the factors on which its period depends?
If the particle starts its motion from mean position, the phase difference between displacement and acceleration is ______.
If the maximum velocity and acceleration of a particle executing SHM are equal in magnitude, the time period will be ______.
A particle executing S.H.M. has a maximum speed of 30 cm/s and a maximum acceleration of 60 cm/s2. The period of oscillation is ______.
A body of mass m is situated in a potential field U(x) = U0 (1 – cos αx) when U0 and α are constants. Find the time period of small oscillations.
A cylindrical log of wood of height h and area of cross-section A floats in water. It is pressed and then released. Show that the log would execute S.H.M. with a time period. `T = 2πsqrt(m/(Apg))` where m is mass of the body and ρ is density of the liquid.
