PHYS 116 Pre-lecture Week 1
The frequency of a physical pendulum depends on which of the following quantities?
- moment of inertia of the physical pendulum - the distance between the pivot and the center of gravity of the physical pendulum
A 5.7-kg mass is attached to one end of a spring with the other end fixed. The mass is pulled 0.35 m from the spring's equilibrium position and released. The mass then moves in simple harmonic motion with a period of 1.2 s. What is the mass's velocity 1.7s after it is released?
-0.92 m/s (v(t)=-wAsin(wt), v(1.7s) = -(5.24 rad/s)(0.35m)sin[(5.24rad/2)(1.7s)])
An object that moves with oscillatory motion has a period of 20.0 s. Determine the frequency of its motion.
0.05 Hz (f = 1/T = 1 / 20s)
A system consists of a block of mass 0.150 kg that is attached to a spring with a spring constant of 200.0 N/m. The block is displaced 5.00 cm from the equilibrium position and it precedes to undergo simple harmonic motion. While the system oscillates, what is the maximum energy in the system?
0.250 J (Umax = (1/2)kA^2, Emax = Umax = (1/2)(200.0 N/m)(5x10^-2 m)^2 = 0.250 J) (1 cm = 0.01 m)
It is known that a damped oscillator has a maximum displacment that is 13.5% of the original amplitude after 0.60 s. What is the value of the time constant?
0.300s (xmax(t) = Ae^-t/T) (ln(xmax(t)/A) = -t/T) (T = -t/(ln(xmax(t)/A)) (T = -0.60s/(ln(0.135))
A 5.7-kg mass is attached to one end of a spring with the other end fixed. The mass is pulled 0.35 m from the spring's equilibrium position and released. The mass then moves in simple harmonic motion with a period of 1.2 s. At what time after it is released will the mass have an acceleration of 6.5 m/s^2?
0.44s (a(t)=-w^2Acos(wt), wt=cos^-1(-a/w^2A), t= cos^-1(-a/w^2A)/w, cos^-1(-(6.5m/s^2)/(5.24rad/s)^2(0.35m))/5.24rad/s)
A simple pendulum is known to have a period of 1.35 s. Determine the length of the pendulum.
0.450 m (T = 2π√L/g) (T/2π = √L/g) (L = T^2g/(2π)^2 (0.450 m = ((1.35s)^2*9.8 m/s^2)/(4π^2))
A physical pendulum, with a mass of 8.3 kg, a length of 0.66 m, and a moment of inertia of 2.1 kg•m?, swings around a pivot at one end and moves back and forth once in 1.4 seconds. How far away from the pivot is the pendulum's center of mass?
0.52 m (w=√(mgd/I) = 2π/T) (mgd/I = ((2π)^2/T^2) (d = I * (2π)^2/T^2 * m * g) (d = (2.1 kg * m * (2π)^2)/((1.4 s)^2 * 8.3 kg * 9.8 m/s^2)
Suppose that we want to consider the oscillating motion of a person's arm as they walk. We will model the arm as a uniform rod that pivots about one of its ends. The moment of inertia of a rod that pivots about one of its ends is given as: I=1/3mL^2 And we will assume the center of gravity of the arm is at the geometric center. What is the frequency of a person's arm that has a length of 0.60 m?
0.790 Hz (ƒ = 1/2π√(mgd/I)) (d=L/2) (ƒ = 1/2π(√(mg(L/2)/(1/3mL^2) (ƒ = 1/2π(√((3/2)(9.8m/s^2/0.60m)) = 0.790 Hz)
Consider Example 14.7 on page 497. The author determines the frequency of the motion to be 2.2 Hz. Suppose that the mass of the block was doubled from 25.0 g to 50.0 g and the spring constant remains at 4.9 N/m. Length = 5.0 cm What would be the new frequency of the motion?
1.60 Hz (F spring = F gravity, k∆x = mg) (k=mg/∆x = (0.050kg)(9.8 m/s^2)/(0.050 m) = 9.8 N/m) (ƒ = 1/2π√(k/m), ƒ = 1/2π√(4.9N/m/0.050kg) = 1.60 Hz)
An object that moves with SHM has a frequency of 0.30 Hz. What is the magnitude of the angular velocity, ω, of this object?
1.9 rad/s (ω=2πf=2π(0.3 s−1)=1.9 rad/s)
A 0.200 kg block is attached to a spring with a spring constant of 100.0 N/m. Determine the value of ΔL, which refers to the stretched length of the spring when the block is in its equilibrium position.
1.96 cm (∆L = mg/k = (0.200 kg)(9.8 m/s^2)/(100.0 N/m) = 0.0196 m) (0.0196m * 100cm/1m)
A 0.200 kg block is attached to a spring with spring constant 100.0 N/m. Determine the value of ΔL, which refers to the stretched length of the spring when the block is in its equilibrium position.
1.96 cm (∆L = mg/k = (0.200kg)(9.81 m/s^2)/(100.0 N/m) = 0.0196 m = 1.96 cm)
A glider is attached to a horizontal spring and oscillates from x = -10.0 cm to x = +10.0 cm. What is the amplitude of the motion?
10.0 cm (It is important to remember that the amplitude is the distance from equilibrium to the maximum, not the distance from the minimum to the maximum.)
A glider is attached to a horizontal spring and oscillates from x = 10.0 cm to x = 10.0 cm. What is amplitude of the motion?
10.0 cm (the amplitude is the distance from the equilibrium to the maximum)
A mass attached to a spring is displaced from its equilibrium position by 5cm and released. The system then oscillates in simple harmonic motion with a period of 1s.If that same mass-spring system is displaced from equilibrium by 10cm instead, what will its period be in this case? 1s 2s 0.5s 1.4s
1s. The period does not depend on the displacement of the mass from the equilibrium position.
A radio wave has a frequency of 5.0 MHz. What is the period of this radiowave?
2.0 x 10^-7 s (T = 1/f = 1 / 5.0 x 10^6 s^-1) (10^6 Hz = 1 megahertz = 1MHz)
Determine the period for a physical pendulum in simple harmonic motion. Mass = 3.8 kg, length = 1.2 m, moment of inertia = 5.9 kg•m?, center of mass = 0.95 m from the axis of rotation.
2.57s (w=√(mgd/I) = 2π/T) (T = 2π√(I/mgd)) T = 2π√((5.9 kg * m^2)/(3.8 kg * 9.8 m/s^2 * 0.95 m))
An object that moves in SHM has a frequency of 3.00 Hz and an amplitude of 15.0 cm. What is the object's maximum speed?
2.83 m/s (vmax=A2πf=Aω=(0.15 m)(2π)(3^s−1)=2.83 m/svmax=A2πf=Aω=(0.15 m)(2π)(3^s−1)=2.83 m/s)
w=
2π/T
Hang on the end of a 5.2-m long rope and swing back and forth in simple harmonic motion. What is the period if you climb halfway up the length of the rope?
3.24 s (w = √(g/1/2L) = 2πƒ = 2π/T) (T = 2π√(1/2L/g) = 2π√((1/2*5.2)/9.8 m/s^2))
Hang on the end of a 5.2-m long rope and swing back and forth in simple harmonic motion. What is the period of this motion?
5.82 s (w = √(g/L) = 2πƒ = 2π/T) (T = 2π√(L/g) = 2π√(5.2/9.8 m/s^2))
The following properties of a mass-on-a-spring system undergoing simple harmonic motion can be changed without affecting the frequency of oscillation?
Amplitude
When the mass that is oscillating is increased, the frequency
Decreases
A toy is pulled down and the spring is stretched. In the instant after the toy is released, the net force on the toy is
Directed upward.
Blocks A and B are attached to separate identical horizontal springs and move with SHM on frictionless horizontal surfaces. Both masses move through the same amplitude, but block A has a larger mass than block B. Is the maximum kinetic energy of block A greater than, less than, or equal to that of block B?
Equal to
The larger the angle is in a pendulum, the __ it will move
Faster
At the equilibrium position
Force of the spring (k*∆x) = force of gravity (mg)
If the frequency of the driver is increased, the amplitude of the oscillation
Increases
The period of a pendulum is __ of the amplitude of oscillation
Independent
The period of a pendulum is __ of the bob's mass.
Independent
An system moves with SHM. When the potential energy of the system is at the maximum value, the kinetic energy of the system:
Is zero.
Two pendulums have identical periods. One has a slightly larger amplitude than the other, but both swing through small angles compared to vertical. Which of the following must be true of the pendulum that has the larger amplitude?
It moves faster at the lowest point in its swing than the other one. (The larger amplitude pendulum has to move faster to complete the cycle in the same amount of time)
If astronauts built a pendulum on the Moon that was exactly the same as one on Earth, what would be different about the pendulum's motion on the Moon?
It would run slower. (The bob of the pendulum would feel less pull because of the weaker gravity on the Moon, so it would not accelerate as quickly.)
You would like a pendulum that swings back and forth once every 2 seconds, but the one you have swings once every 1.9 seconds. Which of the following should you do to adjust it so that it has the desired period?
Make the pendulum slightly longer.
In the motion of a swing at it's lowest point, the potential energy is a zero while the kinetic energy is:
Maximum
You and a friend are on a swing set and her swing is slightly longer than yours. If you both start swinging at the same time, from the same height, where will she be after you have completed one complete swing back and forth?
She will be slightly lower than you but moving upward toward you. (A longer pendulum has a larger period and therefore will gradually run behind a pendulum that is slightly shorter.)
If we double the mass of the pendulum, the frequency
Stays the same
Blocks C and D of identical mass are attached to separate identical springs. Both blocks move with SHM but the amplitude of block C is four times as large as block D. How does the period of block C, TC compare to the period of block D, TD?
TC = TD (It is important to understand that the frequency and period of a simple harmonic oscillator does not depend on the amplitude A. The period of a mass connected to an ideal spring is given as: T = 2π√(m/k)
How is an object's acceleration related to its displacement when it is moving in simple harmonic motion?
The acceleration is proportional to the displacement and points in the opposite direction of the displacement vector
The motor is driving the system at its natural frequency. What will happen to the amplitude of oscillation if the driving frequency is further increased?
The amplitude will decrease
A pendulum clock is made with a metal rod. It keeps perfect time at a temperature of 20°C. At a higher temperature, the metal rod lengthens. How will this change the clock's timekeeping?
The clock will run slow; the dial will be behind the actual time. (A longer rod will increase the period. The clock will tick less often and thus will fall behind.)
If you are driving an oscillatory system at a certain frequency, but the amplitude is much smaller than it could be, you can be certain that
The driving frequency is not matched to the natural frequency of the oscillatory system.
How does increasing the mass of the car affect the frequency of the oscillation?
The increased mass of the car will cause a decrease in the frequency of the oscillation
Which vani bee will change the amount of time it takes the pendulum to swing back and forth?
The length! L
When the marble is at rest in its equilibrium position, what is the net force on the marble?
The net force is zero (no acceleration)
When the marble is to the left of equilibrium position, what is the direction of the restoring force?
The restoring force points down and to the right
When a glider is displaced to the right of the equilibrium position, what is the direction of the restoring force on the glider?
The restoring force points to the left
When the glider is displaced to the right of the equilibrium position, what is the direction of the restoring force on the glider?
The restoring force points to the left.
A simple pendulum consists of a string of length L that is attached to a ball of mass m. The pendulum has a period T. If the mass of the ball is doubled, the new period will be:
Tnew = T (The period of a simple pendulum is independent of the mass of the bob.)
A simple pendulum consists of a string of length L that is attached to a ball of mass m. The pendulum has a period T. If the length of the string is doubled, the new period will be:
Tnew = √2T
At this point, the toy is moving down and passing through its equilibrium position. The toy's acceleration at this point is
Zero
In the motion of a swing at it's highest point, the potential energy is a maximum while the kinetic energy is:
Zero
When the toy is at the highest point its motion, its velocity is __ and its acceleration is __
Zero, negative
At this point, the toy is at its lowest position and the spring is at its maximum extension. The toy's velocity is __ and it's acceleration is __
Zero, positive (at zero because the toy cannot move any lower, and the acceleration is positive because the net force is positive going upward)
Which of the following equations describes the magnitude of the maximum acceleration of an object in SHM?
amax=A(ω)^2
When a car drives over a speed bump and oscillates up and down in simple harmonic motion, at which position during the motion is the acceleration of the car the greatest?
at the maximum amplitude, x = A
Two oscillating systems have periods T1 and T2, with T1 < T2. How are the frequencies of the two systems related?
f1 > f2 ( frequency is the inverse of the period, so a shorter period implies a larger frequency)
Four mass-spring systems have masses and spring constants shown here. Rank in order, from highest to lowest, the frequencies of the oscillations. A. Spring constant = 1k, Mass: 4m B. Spring constant = 1/2k, Mass: m C. Spring constant 1k, Mass: 2m D. Spring constant 2k, Mass: m
fD > fB = fC > fA (The frequency is determined by the ratio of the spring constant to the mass.)
While on Mars, hang on the end of a 5.2-m long rope and swing back and forth in simple harmonic motion. The period of your motion is 7.44 s. What is the acceleration due to gravity on Mars?
gMars = 3.71 m/s^2 (T = 2π√(L/g)) (T^2/(2π)^2 = L/g) (g = L/T^2/(2π)^2) (g = 5.2/((7.44)^2/((2π)^2)))
Consider a bob of mass mm attached to the end of a string of length LL. The bob is set in motion. The tension force exerted on the bob:
has only a centripetal component
The driving frequency is __________________ of the natural frequency.
independent
When the mass is at y = +A, the acceleration of the mass:
is at its maximum negative acceleration.
When the mass passes through the equilibrium position:
it is moving at its maximum speed but it is not possible to tell the direction without more information.
Frequency and period depend on the
length, L
For a system that has a __ restoring force, __ force on the __ object is toward the __ position and is __ to the distance from the equilibrium
linear, net, oscillating, equilibrium, proportional
a long pendulum has a ____ period & a short pendulum has a _____ period.
longer, shorter
At equilibrium, the velocity is at its __ value
maximum
The phenomenon that occurs when the driving frequency matches the natural frequency is called ______________.
resonance
A pendulum that must swing a large distance must do so in the same amount of time as one that swings a __ distance, that is if they both have the sane length and therefore, the sane period.
smaller
low frequency
soft spring, high mass
Doubling the length makes the period __ times larger
sqrt(2)
high frequency
stiff spring, low mass
ax =
x-g/L
sin(theta)
x/L
Determine the location(s) of the mass when the velocity of the mass is zero.
y = -A, y= +A
