PHY 141 Assignment 7 Summer 2018

Reading: Fluids, Oscillations.

Key concepts: Pressure, velocity field, buoyancy, Bernoulli’s theorem; periodic motion, simple harmonic motion, damped and driven oscillations.

1. The king ordered a crown of solid pure gold, but what he got didn’t feel right, so he asked Archimedes to verify that it was really solid and all gold. Archimedes realized he needed a way to measure the density of the crown, but he couldn’t damage it. There are various stories about what he did. The most likely is this.

He balanced the crown with an equal weight of pure gold as shown. Then he lowered the whole system slowly into water and watched to see if the items were still balanced.

How did this allow him to decide whether the king had been cheated? [He had been.]

2. Questions about hydrostatics.

a. The teapot effect. It is well known that if you pour a liquid too slowly from a spout it will double back under the spout and may miss the container you are pouring into. In the drawing fluid comes slowly out of the opening and runs back underneath the pipe. Adhesion makes it turn the corner, but what holds it against the bottom of the pipe for such a distance? [Consider the pressure at the top and bottom of the stream at point A.]

b. The giraffe’s head is 2 m above its heart, which is 2 m above the ground. Discuss and explain two things about this animal: (a) The skin around its lower legs is very tight and thick; (b) When it drinks from a source at ground level it slowly spreads it legs before lowering its head.

3. A “Cartesian diver” is a small glass cylinder open at the bottom which has a pocket of air trapped inside. It is in a sealed flexible plastic container nearly filled with water; initially the diver is floating at the surface as shown. When one squeezes the sides of the container, reducing the volume of the air above the water, the diver sinks toward the bottom. When the squeezing force is removed the diver rises again and floats. Explain this behavior in detail. [What happens to the average density of the diver, and why?]

crown gold

Reading: Fluids, Oscillations.

Key concepts: Pressure, velocity field, buoyancy, Bernoulli’s theorem; periodic motion, simple harmonic motion, damped and driven oscillations.

1.! The king ordered a crown of solid pure gold, but what he got didn’t feel right, so he asked Archimedes to verify that it was really solid and all gold. Archimedes realized he needed a way to measure the density of the crown, but he couldn’t damage it. There are various stories about what he did. The most likely is this.

! He balanced the crown with an equal weight of pure gold as shown. Then he lowered the whole system slowly into water and watched to see if the items were still balanced.

! How did this allow him to make a determination of whether the king had been cheated? [He had been.]

2.! Questions about hydrostatics.

a.! The teapot effect. It is well known that if you pour a liquid too slowly from a spout it will double back under the spout and may miss the container you are pouring into. In the drawing fluid comes slowly out of the opening and runs back underneath the pipe. Adhesion makes it turn the corner, but what holds it against the bottom of the pipe for such a distance? [Consider the pressure at the top and bottom of the stream at point A.]

b.! The giraffe’s head is 2 m above its heart, which is 2 m above the ground. Discuss and explain two things about this animal: (a) The skin around its lower legs is very tight and thick; (b) When it drinks from a source at ground level it slowly spreads it legs before lowering its head.

3.! A “Cartesian diver” is a small glass cylinder open at the bottom which has a pocket of air trapped inside. It is in a sealed flexible plastic container nearly filled with water; initially the diver is floating at the surface as shown. When one squeezes the sides of the container, reducing the volume of the air above the water, the diver sinks toward the bottom. When the squeezing force is removed the diver rises again and floats. Explain this behavior in detail. [What happens to the average density of the diver, and why?]

crown gold

a

PHY 141! Assignment 7! Summer 2018

PHY 141 Assignment 7 Summer 2018

4. A hydraulic jack consists of a cylinder (1) with a small cross section area a, connected by a pipe to a cylinder (2) with larger cross section A. Each cylinder is fitted with a piston that can move vertically, and the system is filled with an incompressible fluid. A heavy load to be raised rests on the piston in 2; the total mass of the load and the piston is M. A force is exerted downward on the piston in 1, resulting (by Pascal’s law) in an upward force on the piston in 2. Comment on the validity of the following statements.

a. The pressure in both cylinders is the same.

b. As the piston in 1 moves down distance d the piston in 2 moves up distance .

c. To lift the piston in 2 and the load at constant speed, .

d. The work done by is equal to the work done by .

5. An aspirator mixes small amounts of a fluid (e.g., liquid fertilizer) with water so it can be sprayed over a large area. As shown, water from a hose passes through a constriction of cross-section a, where a vertical tube introduces the fluid (moving upward at negligible speed) into the rapidly moving water. The mixture then flows out at speed through an opening of larger cross-section A. The height through which the fluid must rise is h. Assume the fluid and water have the same density , and find the minimum ratio of the two cross-sections. [Find the water pressure in the constriction.] Ans:

.

6. Questions about the Bernoulli effect in air.

a. You can lift a card off a table by blowing horizontally over the top of the card. Explain.

b. In a hurricane a flat roof is removed from a building by a wind that is blowing mostly horizontally. Explain.

c. Race cars are designed to direct air striking the front of the car to go underneath it. Why?

F1F2

d ⋅(a/A)

F1 =Mg ⋅(A/a)

F1 F2

v0

ρ A/a

(A/a)2 = 1+ 2gh/v02

4.! A hydraulic jack consists of a cylinder (1) with a small cross section area a, connected by a pipe to a cylinder (2) with larger cross section A. Each cylinder is fitted with a piston that can move vertically, and the system is filled with an incompressible fluid. A heavy load to be raised rests on the piston in 2; the total mass of the load and the piston is M. A force F1

is exerted downward on the piston in 1, resulting in an upward force F2 on the

piston in 2. Comment on the validity of the following statements.

a.! The pressure in both cylinders is the same.

b.! As the piston in 1 moves down distance d the piston in 2 moves up distance d ⋅ (a/A) .

c.! To lift the piston in 2 and the load at constant speed, F1 = Mg ⋅ (A/a) .

d.! The work done by F1 is equal to the work done by F2 .

5.! An aspirator mixes small amounts of a fluid (e.g., liquid fertilizer) with water so it can be sprayed over a large area. As shown, water from a hose passes through a constriction of cross-section a, where a vertical tube introduces the fluid (moving upward at negligible speed) into the rapidly moving water. The mixture then flows out at speed v0 through an

opening of larger cross-section A. The height through which the fluid must rise is h. Assume the fluid and water have the same density ρ , and find the minimum ratio A/a of the two cross-sections.

[Find the water pressure in the constriction.] Ans: (A/a)2 = 1+ 2gh/v02 .

6.! Questions about the Bernoulli effect in air.

a.! You can lift a card off a table by blowing horizontally over the top of the card. Explain.

b.! In a hurricane a flat roof is removed from a building by a wind that is blowing mostly horizontally. Explain.

c.! Race cars are designed to direct air striking the front of the car to go underneath it. Why?

1 2

Supplementary Problems for Topics III

1. An empty steel drum of mass M, height h and cross-section area A is floating partially submerged as shown in water. The bottom of the drum is at depth d below the surface. Express all answers in terms of the given quantities, g, and the density ! of water.

a. What is d?

b. Now an external force of magnitude F is exerted on the top of the drum, pushing it further down into the water by an extra distance x0. Express F in terms of x0 and the other quantities.

c. Prove that if the external force is suddenly removed the drum will execute simple harmonic motion in vertical oscillations. [How does the net force depend on the vertical displacement x of the drum from its equilibrium position?]

d. Find the angular frequency of those oscillations.

2. An aspirator is a device that mixes small amounts of a fluid (such as liquid fertilizer) with water, so that it can be sprayed over a large area. Shown is such a device. Water from a hose (on the left) flows through a constriction, where a vertical tube introduces the fluid (at negligible speed) into the rapid water flow. The height through which the fluid must rise is h as shown. If the fluid-water mixture exits into the air at speed v0, what must be the minimum ratio A/a of the cross-section areas of the two regions? Assume the fluid and water have the same density !. [Find the water pressure in the constriction.]

d

a

h

A v 0 v0

PHY 141! Assignment 7! Summer 2018

Supplementary Problems for Topics III

1. An empty steel drum of mass M, height h and cross-section area A is floating partially submerged as shown in water. The bottom of the drum is at depth d below the surface. Express all answers in terms of the given quantities, g, and the density ! of water.

a. What is d?

b. Now an external force of magnitude F is exerted on the top of the drum, pushing it further down into the water by an extra distance x0. Express F in terms of x0 and the other quantities.

c. Prove that if the external force is suddenly removed the drum will execute simple harmonic motion in vertical oscillations. [How does the net force depend on the vertical displacement x of the drum from its equilibrium position?]

d. Find the angular frequency of those oscillations.

2. An aspirator is a device that mixes small amounts of a fluid (such as liquid fertilizer) with water, so that it can be sprayed over a large area. Shown is such a device. Water from a hose (on the left) flows through a constriction, where a vertical tube introduces the fluid (at negligible speed) into the rapid water flow. The height through which the fluid must rise is h as shown. If the fluid-water mixture exits into the air at speed v0, what must be the minimum ratio A/a of the cross-section areas of the two regions? Assume the fluid and water have the same density !. [Find the water pressure in the constriction.]

d

a

h

A v 0 v0

PHY 141 Assignment 7 Summer 2018

7. An ideal massless spring of stiffness k is attached to a floor and a horizontal plate of mass M. Resting on the plate is a block of mass m.

a. When the system is at rest, how much is the spring compressed?

b. The system is pushed down an extra distance A and released. What is the angular frequency of the vertical oscillation?

c. What is the maximum value of A for which the block will not leave the plate at any point?

8. A spring of stiffness k is attached to a wall and to the frictionless axle of a wheel of mass m, radius R, and moment of inertia about the axle. With the spring stretched distance A and the wheel at rest, the system is released. The floor has sufficient static friction that the wheel rolls without slipping.

a. When the spring is stretched distance x and the wheel’s CM has speed v, what is the total energy of the system? Ans: .

b. What is the maximum speed of the CM? Ans:

c. Show that the motion is SHM and find the angular frequency . Ans: .

9. Questions about oscillations that are not quite simple harmonic.

a. We have been assuming the springs in our examples have no mass. If we took into account the mass of an actual spring attached to a mass m and oscillating, would the real value of be larger or smaller than the one we get from ? Explain.

b. The oscillations of a simple pendulum are SHM in the approximation that , where is the angle made by the string with the vertical.

The approximation for the cosine is more exact if we keep the next term in the series, so . Would this make larger or smaller than ?

ω

I = βmR2

12m(1+ β )v

2 + 12 kx2

vmax2 = kA2

m(1+ β )

ωω 2 = k/m(1+ β )

ωω = k/m

cosθ ≈ 1−θ 2/2 θ

cosθ ≈ 1−θ 2/2+θ 4 /12 ωg/ℓ

Mm

•

PHY 141 Assignment 7 Summer 2018

10. More on small oscillations about equilibrium.

a. The first three terms in a Taylor expansion of a function about are: , where the primes indicate derivatives. Use this to show that the first three terms in the binomial approximation for , where , are:

[Often we keep only the first two.]

b. You will apply this to small oscillations. Let the potential energy of a

system be , where x is positive. Use a graphing

calculator to plot this function for .

c. One sees that there is a minimum at . (You can also verify this by calculus.) Write , and assume . Use the expansion in (a) to write the potential near in terms of z. Ans: .

d. So, apart from a constant, this is quadratic in z. Let it be the potential energy for a particle of mass m. What is the angular frequency of the small oscillations about ? Ans: .

11. Block 1, of mass m, is attached as shown to a spring attached to a wall. The floor is frictionless. Identical block 2, moving as shown with speed collides with block 1 when the spring is at its equilibrium length. The collision is elastic.

a. How are the two blocks moving immediately after the collision?

b. The spring has stiffness k. How far does it compress before block 1 comes momentarily to rest? What happens after that?

12. Consider the same arrangement. Block 1 is oscillating with amplitude and maximum speed . At the instant when the spring is stretched to its maximum, block 2, moving to the left at speed ; strikes block 1 and the two blocks stick together after the brief collision.

a. What is the ratio of the angular frequencies of oscillation after and before the collision?

b. What is the ratio of the amplitudes after and before the collision?

c. What is the ratio of the maximum speeds after and before the collision?

f (x) x = 0f (x) = f (0)+ ′f (0) ⋅x + 12 ′′f (0) ⋅x2 + ...

(1+ x)n x << 1

(1+ x)n = 1+ n ⋅x + 12n(n−1) ⋅x2 + ...

V(x) = −V02x− 1x2

⎛⎝⎜

⎞⎠⎟

V0 = 1

x = 1x = 1+ z z <<1

x = 1 V ≈ −V0(1− 3z2)

ωx = 1 ω = 6V0/m

v0

A0v0

v0

ω/ω0

A/A0v/v0

1 2

PHY 141 Assignment 7 Summer 2018

13. Two blocks, of mass m and 2m, are resting on a frictionless table, attached as shown to an ideal spring of stiffness k. A third block, of mass m, moving to the right with speed as shown, collides with and sticks to the other block of mass m. The combined system then moves to the right.

[Give all answers in terms of m, k, and .]

a. Describe the subsequent motion of the system.

[An animation of this situation is here.]

b. What is the total kinetic energy just after the collision, when the block of mass 2m has not yet started to move? [What is conserved in the collision?] Ans: .

c. What is the kinetic energy at an instant when all the blocks are traveling with the same speed? [This is the kinetic energy of the CM motion alone.] Ans: .

d. The instant in (c) is when the compression or extension of the spring is its maximum amount . What is ? Ans: .

14. Two identical mass-spring systems, A and B, are attached to a flexible horizontal rod as shown. When A is set into oscillation with its natural frequency , the rod begins to vibrate slightly up and down at that frequency. This vibration acts as a driving force for B. We are interested in the average power of the driven oscillation of B, in two cases: (1) B’s natural frequency is ; (2) B’s natural frequency is . Assume A is undamped.

a. Let B have damping such that . What is the ratio of the average power delivered to B in the two

cases? Ans: 17/4.

b. Repeat for the case where . Ans: 226.

c. In case (1) let both systems be undamped. Discuss the energy transfer between the systems over time. (Total energy is conserved.)

[See the notes Oscillations, page 7.]

v0

v0

14mv0

2

18mv0

2

xmax xmax (v0/2) ⋅ m/k

ω0

ω02ω0

b/m =ω0

b/m =ω0/10

A B