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| BETA : Cyclic : Simple Harmonic Motion 01 : Vibration Frequency Period And Amplitude : BETA Physics Homework Exercise
Description : Vibration frequency period and amplitude.
Discussion : Simple harmonic motion (SHM) is periodic with frequency equals the inverse of the period. The amplitude is the maximum positive or negative displacement relative to the mid point or equilibrium position. See Figure Vibration Of A Mass On A Spring
The homework exercise has 2 sets of questions with 4 questions per set.
T, f, c, A
T, omega, dmax, dmin
The basic equations are :
T = 2 π / omega
f = 1 / T
c = (dmax + dmin) / 2
A = (dmax - dmin) / 2
rearranging
omega = 2 π / T = 2 π f
T = 1 / f
f = omega / (2 π)
dmax = c + A
dmin = c - A
where
T = cyclic period
f = frequency
omega = angular velocity (ω)
dmin = minimum displacement
dmax = maximum displacement
c = center or equilibrium displacement
A = amplitude
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| BETA : Cyclic : Simple Harmonic Motion 02 : Simple Harmonic Motion Phasor Diagram : BETA Physics Homework Exercise
Description : Simple harmonic motion phasor diagram.
Discussion : A rotating phasor on the reference circle is often used to represent simple harmonic motion. See Figure Simple Harmonic Motion Phasor Diagram
The homework exercise has 3 sets of questions with 5 questions per set.
omega, theta, y, v, a
theta, omega, a, t, f
f, t, A, y, v
The basic equations are :
omega = 2 π f
theta = omega t
y = A sin(theta)
v = A omega cos(theta)
a = -A omega2 sin(theta) = - omega2 y
rearranging
theta = asin(y / A)
f = omega / (2 π )
t = theta / omega
omega = v / (A cos(theta))
A = -a / (omega2 sin(theta))
where
omega = angular velocity (ω)
theta = phase angle (θ)
y = displacement from equilibrium
v = velocity
a = acceleration
t = time
f = frequency
The angles are in radians. You are recommended to use the trigonometry radian functions sin(), cos() and asin().
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| BETA : Cyclic : Simple Harmonic Motion 03 : Simple Harmonic Motion Displacement Of A Mass And Spring : BETA Physics Homework Exercise
Description : SHM (simple harmonic motion) displacement of a mass and spring.
Discussion : The angular frequency of a vibrating mass and spring is equal to the square root of the spring constant divided by the mass. See Figure SHM Displacement Of A Mass And Spring
The homework exercise has 3 sets of questions with 4 questions per set.
k, omega, T, f
omega, f, k, F
omega, T, m, y
The basic equations are :
k = F / y
omega = √(k / m)
T = 2 π / omega
f = omega / (2 π)
rearranging
omega = 2 π / T
f = omega / (2 π )
k = omega2 m
m = omega2 / k
F = k y
y = F / k
f = 1 / T
T = 1 / f
where
omega = angular velocity (ω)
y = displacement from equilibrium
F = Force
f = frequency
m = mass
T = period
k = spring constant
F is the force required to displace the mass and spring from equilibrium. There is an equal and opposite force exerted by the spring which drives the oscillation.
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| BETA : Cyclic : Simple Harmonic Motion 04 : Simple Harmonic Motion Displacement Of A Pendulum : BETA Physics Homework Exercise
Description : SHM (simple harmonic motion) displacement of a pendulum.
Discussion : When the displacement is small relative to the pendulum length, the angular frequency of a pendulum is approximately equal to the square root of gravity constant divided by the length. See Figure SHM Displacement Of A Pendulum
The homework exercise has 2 sets of questions with 4 questions per set.
F, omega, T, f
f, T, L, m
The basic equations are :
omega = √(g / L)
T = 2 π / omega
f = omega / (2 π)
F = m g y L
rearranging
f = omega / (2 π )
f = 1 / T
T = 1 / f
L = g / omega2
m = F L / (g y)
where
omega = angular velocity (ω)
L = pendulum length
f = frequency
T = period
m = mass
y = displacement from equilibrium
F = force
g = gravity constant
F is the force required to displace the pendulum mass from equilibrium. There is an equal and opposite force exerted by the mass which drives the oscillation.
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| BETA : Cyclic : Simple Harmonic Motion 05 : Simple Harmonic Motion Displacement Of Liquid In A U-Tube : BETA Physics Homework Exercise
Description : SHM (simple harmonic Motion) displacement of liquid in a u-tube
Discussion : The angular frequency is equal to the square root of the displaced fluid weight divided by the toal fluid mass. See Figure SHM Displacement Of Liquid In A U-Tube
The homework exercise has 2 sets of questions with 4 questions per set.
F, omega, T, f
m, y, T, f
The basic equations are :
F = 2 rho X g y
omega = √(2 rho X g / m)
T = 2 π / omega
f = omega / (2 π)
rearranging
m = 2 rho X g / omega2
y = F / (2 rho X g)
f = 1 / T
T = 1 / f
where
omega = angular velocity (ω)
rho = liquid density (ρ)
f = frequency
T = period
m = mass
y = displacement from equilibrium
F = force
g = gravity constant
X = liquid cross section area
F is the force required to displace the fluid from equilibrium. There is an equal and opposite force exerted by the fluid which drives the oscillation.
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| BETA : Cyclic : Simple Harmonic Motion 11 : Simple Harmonic Motion Energy Of A Mass And Spring : BETA Physics Homework Exercise
Description : Simple harmonic motion energy of a mass and spring.
Discussion : An oscillating mass spring system has constant energy (ignoring friction effects). The energy changes between stored spring energy and kinetic energy, but the total energy remains constant. The velocity v and the kinetic energy Ek are measured at the displacement y. See Figure SHM Displacement Of A Mass And Spring
The homework exercise has 3 sets of questions with 4 questions per set.
Et, vmax, Ek, v
Ek, y, vmax, A
m, Et, k, A
The basic equations are :
Et = k / 2 A2
vmax = √(2 Et / m)
Ek = Et - k / 2 y2
v = √(2 Ek / m);
rearranging
Et = m / 2 vmax2
Ek = m / 2 v2
y = √(2 (Et - Ek) / k)
A = √(2 Et / k)
m = 2 Ek / v2
k = 2 (Et - Ek) / y2
where
A = amplitude
y = displacement from equilibrium
m = mass
k = spring constant
Et = total energy
Ek = kinetic energy
vmax = maximum velocity
v = velocity
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| BETA : Cyclic : Simple Harmonic Motion 12 : Simple Harmonic Motion Energy Of A Pendulum : BETA Physics Homework Exercise
Description : Simple harmonic motion energy of a pendulum.
Discussion : An oscillating pendulum has constant energy (ignoring friction effects). The energy changes between stored potential energy and kinetic energy, but the total energy remains constant. The velocity v and the kinetic energy Ek are measured at the displacement y. The pendulum equations are only valid where A < 0.1 L. See Figure SHM Displacement Of A Pendulum
The homework exercise has 3 sets of questions with 4 questions per set.
Et, vmax, Ek, v
Ek, y, vmax, A
m, Et, L, A
The basic equations are :
Et = m g A2 / L
vmax = √(2 Et / m)
Ek = Et - m g y2 / L
v = √(2 Ek / m);
rearranging
Et = m / 2 vmax2
Ek = m / 2 v2
y = √(L (Et - Ek) / (m g))
A = √(L Et / (m g))
m = 2 Ek / v2
L = m g y2 / Ek
where
A = amplitude
y = displacement from equilibrium
m = mass
g = gravity constant
Et = total energy
Ek = kinetic energy
vmax = maximum velocity
v = velocity
L = pendulum length
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| BETA : Cyclic : Simple Harmonic Motion 13 : Simple Harmonic Motion Energy Of Liquid In A U-Tube : BETA Physics Homework Exercise
Description : Simple harmonic energy of liquid in a u-tube.
Discussion : The oscillating fluid has constant energy (ignoring friction effects). The energy changes between stored potential energy and kinetic energy, but the total energy remains constant. The velocity v and the kinetic energy Ek are measured at the displacement y. See Figure SHM Displacement Of Liquid In A U-Tube
The homework exercise has 3 sets of questions with 4 questions per set.
Et, vmax, Ek, v
Ek, y, vmax, A
m, Et, rho, A
The basic equations are :
Et = rho X g A2
vmax = √(2 Et / m)
Ek = Et - rho X g y2
v = √(2 Ek / m);
rearranging
Et = m / 2 vmax2
Ek = m / 2 v2
y = √((Et - Ek) / (rho X g))
A = √(Et / (rho X G))
m = 2 Ek / v2
rho = (Et - Ek) / (X G y2)
where
A = amplitude
y = displacement from equilibrium
m = mass
g = gravity constant
Et = total energy
Ek = kinetic energy
vmax = maximum velocity
v = velocity
rho = liquid density (ρ)
X = u-tube cross section area
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