Simple Harmonic Motion Harmonic motion due to a net restoring force directly proportional to the displacement Eample: Spring motion: F = -k Net Force: d! k = m dt " F = ma d dt + k m = 0 Equation of motion for the simple harmonic oscillator
Simple Harmonic Motion = Acos(! t + ") Net Force: d dt d! k = m dt " k + = 0 m F = ma d d = ( A cos(! t + ")) = # A! sin(! t + ") dt dt d d d d = = # A dt dt dt dt t + = #!! + " [! sin(! ")] Acos( t ) Now substitute in the SHM equation: d k + 0 = dt m k #! Acos(! t + ") + Acos(! t + ") = 0 m k ( #! ) Acos(! t + ") = 0 m $! = k m
Angular Frequency!:! = k m Frequency, f :! = " f Period, T: "! = T
1. Nikita devised the following method of measuring the muzzle velocity of a rifle. She fires a bullet into a wooden block (mass M) resting on a smooth surface, and attached to a spring with a spring constant k. The bullet, whose mass is m, remains embedded in the wooden block. She measures the distance that the block recoils and compresses the spring to be A. What is the speed of the bullet? 1) v = ) v = 3) v = 4) v = m + M m m M ka ka ( m + M ) ka M m M ka
. The block is going to eecute simple harmonic motion. What is the frequency of oscillation? 1) 1 " ) 1 " 3) 1 " k m + M k M k m 4) 1 ka " m + M 5) None of the above
3. A mass (m) is connected to two springs with spring constants k 1 and k. The mass is displaced from the equilibrium position and released. (a) Show that the mass eecutes simple harmonic motion. (b) Find the frequency of oscillation.
3. A mass (m) is connected to two springs with spring constants k 1 and k. The mass is displaced from the equilibrium position and released. (a) Show that the mass eecutes simple harmonic motion. (b) Find the frequency of oscillation. a) To show that it eecutes SHM, we need the restoring force equation.
3. A mass (m) is connected to two springs with spring constants k 1 and k. The mass is displaced from the equilibrium position and released. (a) Show that the mass eecutes simple harmonic motion. (b) Find the frequency of oscillation. a) To show that it eecutes SHM, we need the restoring force equation. If we move the mass to the right by an amount, then the restoring force is
3. A mass (m) is connected to two springs with spring constants k 1 and k. The mass is displaced from the equilibrium position and released. (a) Show that the mass eecutes simple harmonic motion. (b) Find the frequency of oscillation. a) To show that it eecutes SHM, we need the restoring force equation. Spring 1 Spring F 1 = "k 1 F = "k If we move the mass to the right by an amount, then the restoring force is
3. A mass (m) is connected to two springs with spring constants k 1 and k. The mass is displaced from the equilibrium position and released. (a) Show that the mass eecutes simple harmonic motion. (b) Find the frequency of oscillation. a) To show that it eecutes SHM, we need the restoring force equation. Spring 1 Spring F 1 = "k 1 F = "k If we move the mass to the right by an amount, then the restoring force is F = F 1 + F = "(k 1 + k ) = "k eff
3. A mass (m) is connected to two springs with spring constants k 1 and k. The mass is displaced from the equilibrium position and released. (a) Show that the mass eecutes simple harmonic motion. (b) Find the frequency of oscillation. a) To show that it eecutes SHM, we need the restoring force equation. Spring 1 Spring F 1 = "k 1 F = "k If we move the mass to the right by an amount, then the restoring force is F = F 1 + F = "(k 1 + k ) = "k eff k effective = k 1 + k f = " k 1 + k m
4. A mass (m) is connected to two springs with spring constants k 1 and k. The mass is displaced from the equilibrium position and released. (a) Show that the mass eecutes simple harmonic motion. (b) Find the frequency of oscillation.
4. A mass (m) is connected to two springs with spring constants k 1 and k. The mass is displaced from the equilibrium position and released. (a) Show that the mass eecutes simple harmonic motion. (b) Find the frequency of oscillation. If we apply a force F to stretch the springs, then the total displacement " = " 1 + " Δ 1 Δ
4. A mass (m) is connected to two springs with spring constants k 1 and k. The mass is displaced from the equilibrium position and released. (a) Show that the mass eecutes simple harmonic motion. (b) Find the frequency of oscillation. If we apply a force F to stretch the springs, then the total displacement " = " 1 + " Δ 1 Δ = # F k 1 # F k F = F 1 = F
4. A mass (m) is connected to two springs with spring constants k 1 and k. The mass is displaced from the equilibrium position and released. (a) Show that the mass eecutes simple harmonic motion. (b) Find the frequency of oscillation. If we apply a force F to stretch the springs, then the total displacement " = " 1 + " Δ 1 Δ F = F 1 = F = # F # F k 1 k $ = # F& 1 + 1 % k 1 k = # F k eff ' ) (
4. A mass (m) is connected to two springs with spring constants k 1 and k. The mass is displaced from the equilibrium position and released. (a) Show that the mass eecutes simple harmonic motion. (b) Find the frequency of oscillation. If we apply a force F to stretch the springs, then the total displacement " = " 1 + " Δ 1 Δ F = F 1 = F 1 = 1 + 1 k eff k 1 k = # F # F k 1 k $ = # F& 1 + 1 % k 1 k = # F k eff ' ) (
5. A mass (m) is supported by two springs with spring constants k 1 and k. The mass is displaced from the equilibrium position and released. Find the frequency of oscillation.
5. A mass (m) is supported by two springs with spring constants k 1 and k. The mass is displaced from the equilibrium position and released. Find the frequency of oscillation. When the mass is pulled down by a distance, F = F 1 + F = "k 1 " k
5. A mass (m) is supported by two springs with spring constants k 1 and k. The mass is displaced from the equilibrium position and released. Find the frequency of oscillation. When the mass is pulled down by a distance, F = F 1 + F = "k 1 " k = "(k 1 + k ) = "k eff
6. A uniform stick of length L and mass M is pivoted on a hinge at one end and held horizontal by a spring with a spring constant k. If the stick oscillates up and down slightly, then show its motion is simple harmonic. What is its frequency? L
6. A uniform stick of length L and mass M is pivoted on a hinge at one end and held horizontal by a spring with a spring constant k. If the stick oscillates up and down slightly, then show its motion is simple harmonic. What is its frequency? In equilibrium: Stick: horizontal position Spring: etended by 0 L
6. A uniform stick of length L and mass M is pivoted on a hinge at one end and held horizontal by a spring with a spring constant k. If the stick oscillates up and down slightly, then show its motion is simple harmonic. What is its frequency? L In equilibrium: Stick: horizontal position Spring: etended by 0 # " = Mg $ & l ' % ) * k ( 0 l = 0
6. A uniform stick of length L and mass M is pivoted on a hinge at one end and held horizontal by a spring with a spring constant k. If the stick oscillates up and down slightly, then show its motion is simple harmonic. What is its frequency? θ L In equilibrium: Stick: horizontal position Spring: etended by 0 # " = Mg $ & l ' % ) * k ( 0 l = 0 If the stick is displaced through a small angle θ, and the spring is etended by an amount # " = Mg $ & l ' % ) * k( + ( 0 )l = I+
6. A uniform stick of length L and mass M is pivoted on a hinge at one end and held horizontal by a spring with a spring constant k. If the stick oscillates up and down slightly, then show its motion is simple harmonic. What is its frequency? θ L In equilibrium: Stick: horizontal position Spring: etended by 0 # " = Mg $ & l ' % ) * k ( 0 l = 0 If the stick is displaced through a small angle θ, and the spring is etended by an amount # " = Mg $ & l ' % ) * k( + ( 0 )l = I+ * kl = I d, dt * kl, = I d, dt
6. A uniform stick of length L and mass M is pivoted on a hinge at one end and held horizontal by a spring with a spring constant k. If the stick oscillates up and down slightly, then show its motion is simple harmonic. What is its frequency? θ d " dt + kl I " = 0 # = kl I = L kl Ml /3 = 3k M In equilibrium: Stick: horizontal position Spring: etended by 0 # " = Mg $ & l ' % ) * k ( 0 l = 0 If the stick is displaced through a small angle θ, and the spring is etended by an amount # " = Mg $ & l ' % ) * k( + ( 0 )l = I+ * kl = I d, dt * kl, = I d, dt
Force between two molecules F (Positive) Repulsive force (Negative) Attractive force r Simple Harmonic Motion
Problem 7. In some diatomic molecules, the force each atom eerts on the other can be approimated by C =! + r 3, where r is the atomic separation, and C and D are positive constants. (a) Graph F versus r from r = 0 to r = D/C. F D r F (Positive) Repulsive force (Negative) Attractive force r Simple Harmonic Motion
Problem 7. In some diatomic molecules, the force each atom eerts on the other can be approimated by C =! + r 3, where r is the atomic separation, and C and D are positive constants. (a) Graph F versus r from r = 0 to r = D/C. (b) Find the equilibrium position r 0 in terms of the constants C and D. F F D r (Positive) Repulsive force (Negative) Attractive force r Simple Harmonic Motion
Problem 7. In some diatomic molecules, the force each atom eerts on the other can be approimated by C =! + r 3, where r is the atomic separation, and C and D are positive constants. (a) Graph F versus r from r = 0 to r = D/C. (b) Find the equilibrium position r 0 in terms of the constants C and D. F D r (Positive) Repulsive force (Negative) Attractive force F r F(r 0 ) = " C r 0 + D r 0 3 = 0 r 0 = D C Simple Harmonic Motion
Problem 7. In some diatomic molecules, the force each atom eerts on the other can be approimated by F C =! + r 3, where r is the atomic separation, and C and D are positive constants. (a) Graph F versus r from r = 0 to r = D/C. (b) Find the equilibrium position r 0 in terms of the constants C and D. (c) Assuming that, for small displacements Δr << r 0 the motion is approimately simple harmonic, determine the force constant k in terms of C and D. r 0 = D C F(r o + "r) = # = # = # D r C (r 0 + "r) + D (r 0 + "r) 3 1 (r 0 + "r) 3 [Cr 0 + C"r # D] C"r (r 0 + "r) 3 $ # C"r (r 0 ) 3 = #k eff "r
8. A cylindrical block of wood of mass m and cross-sectional area A and height h is floating in water. You give it a nudge from the top, and the block starts to oscillate in water. The water density is ρ w and acceleration due to gravity is g. Show that it eecutes Simple Harmonic Motion. Find the effective spring constant. In equilibrium, F buoyancy = mg F buoyancy mg
8. A cylindrical block of wood of mass m and cross-sectional area A and height h is floating in water. You give it a nudge from the top, and the block starts to oscillate in water. The water density is ρ w and acceleration due to gravity is g. Show that it eecutes Simple Harmonic Motion. Find the effective spring constant. In equilibrium, F buoyancy = mg F buoyancy When you push the block into the water (by Δ), there is an additional buoyant force, equal to the additional water displaced. F net = "# water ga$ mg
8. A cylindrical block of wood of mass m and cross-sectional area A and height h is floating in water. You give it a nudge from the top, and the block starts to oscillate in water. The water density is ρ w and acceleration due to gravity is g. Show that it eecutes Simple Harmonic Motion. Find the effective spring constant. In equilibrium, F buoyancy = mg F buoyancy When you push the block into the water (by Δ), there is an additional buoyant force, equal to the additional water displaced. F net = "# water ga$ mg k effective = " water ga
Problem 9. In Alice in Wonderland, Alice imagines a 10-cm diameter hole drilled all the way through the center of the earth. Standing on one end of the hole, she drops an apple through to see that the apple makes simple harmonic motion about the center of the earth. (a) At a point from the center, what is the gravitational force eperienced by the apple of mass m? Assume that the earth has a uniform mass density ρ. Indicate both magnitude and sign. (Hint: The apple at a distance from the center eperiences the gravitational attraction only from the part of the earth within a sphere of radius.) (b) What is the effective spring constant k? (c) How long will the apple take to return to Alice, if all frictional effects are ignored?
Problem 9. (a) At a point from the center, what is the gravitational force eperienced by the apple of mass m? Assume that the earth has a uniform mass density ρ. Indicate both magnitude and sign. (Hint: The apple at a distance from the center eperiences the gravitational attraction only from the part of the earth within a sphere of radius.) r E
Problem 9. (a) At a point from the center, what is the gravitational force eperienced by the apple of mass m? Assume that the earth has a uniform mass density ρ. Indicate both magnitude and sign. (Hint: The apple at a distance from the center eperiences the gravitational attraction only from the part of the earth within a sphere of radius.) r E
Problem 9. (a) At a point from the center, what is the gravitational force eperienced by the apple of mass m? Assume that the earth has a uniform mass density ρ. Indicate both magnitude and sign. (Hint: The apple at a distance from the center eperiences the gravitational attraction only from the part of the earth within a sphere of radius.) r E $ M M* = "V = E ' & % 4#r 3 E /3( ) $ 4# 3 ' & 3 ) = M E 3 3 % ( r E
Problem 9. (a) At a point from the center, what is the gravitational force eperienced by the apple of mass m? Assume that the earth has a uniform mass density ρ. Indicate both magnitude and sign. (Hint: The apple at a distance from the center eperiences the gravitational attraction only from the part of the earth within a sphere of radius.) r E F = " GmM * = " GmM E r E 3 3 = " GmM E r E 3 $ M M* = "V = E ' & % 4#r 3 E /3( ) $ 4# 3 ' & 3 ) = M E 3 3 % ( r E
Problem 9. (a) At a point from the center, what is the gravitational force eperienced by the apple of mass m? Assume that the earth has a uniform mass density ρ. Indicate both magnitude and sign. (Hint: The apple at a distance from the center eperiences the gravitational attraction only from the part of the earth within a sphere of radius.) r E F = " GmM * = " GmM E r E 3 3 = " GmM E r E 3 $ M M* = "V = E ' & % 4#r 3 E /3( ) $ 4# 3 ' & 3 ) = M E 3 3 % ( r E k eff = GmM E r E 3
A Mylar balloon (of mass m when empty) is filled with helium and occupies a volume V. The balloon is tied up to a long string of length l, as shown in the figure. When the balloon is pushed sideways, it starts to oscillate left and right in simple harmonic motion like an up-side-down pendulum. Air He inside