physics problems for Circular motion
What is the angular speed of the tip of the minute hand on a clock, in rad/s?
1 hr x 60min x 60 sec = 3600 sec 2π / 3600 = 0.0017 rad/sec
what is the maximum flow rate of water for laminar flow in a 5.0 cm diameter pipe?
2000 = PVR/u = 1000(V)(0.05) / 1 x 10^-3 V= 1x10^-3 (2000) / 1000(0.05) = 0.04 m/s Q=AV = π(P/𝛿)^2 V Q= π(0.025)^2(0.04) = 7.85 x 10^-5
As a roller coaster car crosses the top of a 40-m-diameter loop-the-loop, its apparent weight is the same as its true weight. What is the car's speed at the top?
2rg = v^2 solve for v^2 2(40/2)(9.81)=v^2 square root v= 19.81 m/s
A 1.0-cm-diameter pipe widens to 2.0 cm, then narrows to 0.50 cm. Liquid flows through the first segment at a speed of 4.0 m/s. A) What are the speeds in the second and third segments? B) What is the volume flow rate through the pipe?
A) A1V1 = A2V2 --> V2=A1V1/A2 area for 1st segment A1=π(d1/2)^2 = π(1.0/2)^2 = 7.9 x 10^-5 for 2nd segment A2=π(0.02/2)^2 = 3.14 x 10^-4 V2= A1V1/A2 = (7.9x10^-5)(4.0)/(3.14x10^-4) = 0.99 ~ 1.0 m/s for 3rd segment A3= π(0.03/2)^2 = 7.06 x 10^-4 V3= A1V1(A2V2)/A3 = (7.9x10^-5)(4.0)(3.14x10^-4)(1.0)/7.06x10^-4 =1.41 x10^-4 B) Q=A1V1 = (7.9 x 10^-5)(4.0) =3.16 x 10^-4 (1000L) = 0.316 L/s
We can model a pine tree in the forest as having a compact canopy at the top of a relatively bare trunk. Wind blowing on the top of the tree exerts a horizontal force, and thus a torque that can topple the tree if there is no opposing torque. Suppose a tree's canopy presents an area of 9.0 to the wind centered at a height of 7.0 m above the ground. (These are reasonable values for forest trees.) If the wind blows at 6.5 m/s: A) What is the magnitude of the drag force of the wind on the canopy? Assume a drag coefficient of 0.50. B)What torque does this force exert on the tree, measured about the point where the trunk meets the ground?
A) Cd= 0.5 p=1.2 A= 9 F= Cd(A)(p)(v^2) / 2 F= (0.5)(9.0)(1.2)(6.5)^2 = 114.07 N B) τ = Fr = Frsinθ τ= (114.07)(7sin90) τ= 798.49 Nm
An athlete at the gym holds a 3.0 kg steel ball in his hand. His arm is 70 cm long and has a mass of 4.0 kg. What is the magnitude of the gravitational torque about his shoulder if he holds his arm A) Straight out to his side, parallel to the floor? B) Straight, but 45° below horizontal?
A) Tg = Ts + Ta Tg = (3.0)(9.81)(0.7) + (4.0)(9.81)(0.7)(0.3) Tg= 28.84 Nm B) Tg= Ts + Ta = ms x g x 0.7cos45 + ma x g x 0.7cos45 x 0.3 Tg= 3.0(9.81)(0.7)(cos45) + (4.0)(9.81)(0.7)(cos45)(0.3) Tg= 20.39 Nm
During typical urination, a man releases about 400 mL of urine in about 30 seconds through the urethra, which we can model as a tube 4 mm in diameter and 20 cm long. Assume that urine has the same density as water, and that viscosity can be ignored for this flow. A) What is the flow speed in the urethra? B) If we assume that the fluid is released at the same height as the bladder and that the fluid is at rest in the bladder, what gauge pressure in the bladder would be necessary to produce this flow?
A) UA = 400ml/30sec A= π (4/2mm)^2 = 4 x 10^-6 π 1 ml= 10^-6 m^3 --> 400mL = 400 x 10^-6 m3 400 x 10^-6 / 30 sec = U (4x10^-6 π) U= 400 / 30(4π) = 1.06 m/s B) Pi= (p/δ)Ui^2 = pf + p/δUf^2 bladder pressure = Pi-Pf = p/δUf^2 = 1000/2 (1.06)^2 = 560 N/m^2
Modern wind turbines are larger than they appear, and despite their apparently lazy motion, the speed of the blades tips can be quite high—many times higher than the wind speed. A typical modern turbine has blades 56 m long that spin at 13 rpm A) what is the speed of the blade? B) what is the centripetal acceleration?
A) V=rw w= rpm = 13 (2π/60) = 1.36 V= (56)(1.36) = 76.2 m/s B) a=rw^2 (56)(1.36)^2 = 103.6 m/s^2
A fast pitch softball player does a "windmill" pitch, illustrated in Figure P6.18, moving her hand through a circular arc to pitch a ball at 70 mph. The 0.19 kg ball is 50 cm from the pivot point at her shoulder. At the lowest point of the circle, the ball has reached its maximum speed. A) At the bottom of the circle, just before the ball leaves her hand, what is its centripetal acceleration? B) What are the magnitude and direction of the force her hand exerts on the ball at this point?
A) a=v^2/r 70 mph --> 31.29 m/s 50 cm --> 0.5 m a= (31.29)^2 / (0.5) = 1958.12 m/s^2 B) F=ma F= (0.19)(1958.12) = 372.0 N
The California sea lion is capable of making extremely fast, tight turns while swimming underwater. In one study, scientists observed a sea lion making a circular turn with a radius of 0.35 m while swimming at 4.2 m/s. A) what is the sea lions centripetal acceleration? B) what % is the acceleration of that of an F-15 jet's max centripetal acceleration of 9g?
A) a=v^2/rg a= (4.2)^2 / (0.35)(9.81) = 5.14 g B) (5.14 / 9 ) x (100%) = 57.1 %
The hammer throw was one of the earliest Olympic events. The last complete rotation from the olympic winner turn took only 0.43 s. The radius of the ball's path, including her extended arms, was 2.1 m. A) What was the frequency of this rotation? B) What was the speed of the ball? C) What was the ball's acceleration, in units of g?
A) f=1/T f= 1/0.43s = 2.33 hz B) V= 2πfr V= 2π(2.33)(2.1) = 30.7 m/s C) a=v^2 / r (30.7)^2 / 2.1 = 448.8 m/s^2 / 9.81 = 45.74 g
The 1.00-cm-long second hand on a watch rotates smoothly. A) What is its angular velocity? B) What is the speed of the tip of the hand?
A) time period = 60 sec w=2π/60 = 0.104 rad/sec speed = rw = 1.00(0.104) = 0.1047 cm/sec B) v=rw v= (0.01)(0.1047) = 0.001047 m/s
In the very Dutch sport of Fierljeppen, athletes run up to a long pole and then use it to vault across a canal. At the very top of his arc, a 55 kg vaulter is moving at 2.5 m/s and is 5.1 m from the bottom end of the pole. What vertical force does the pole exert on the vaulter?
F=mg - mv^2 / r F= (55)(9.81) - (55)(2.5)^2 / (5.1) = 472.15 N
A regulation table tennis ball is a thin spherical shell 40 mm in diameter with a mass of 2.7 g. What is its moment of inertia about an axis that passes through its center?
I = 2/3 mr^2 m=2.7 r=40/2=20mm I= 2/3(2.7)(20)^2 I=720 ~ 7.2 x 10^2
A small grinding wheel has a moment of inertia of 4.0 x 10^-5 What net torque must be applied to the wheel for its angular acceleration to be 150 rad/s^2?
I = 4.0 x 10^-5 a= 150 rad/s^2 τ = (4.0 x 10^-5)(150) = 6.0 x 10^-3 Nm
To keep blood from pooling in their lower legs on plane trips, some people wear compression socks. These socks are sold by the pressure they apply; a typical rating is 20 mm Hg. Over what vertical distance can this pressure move the blood?
P=hpg ΔPsock = ΔPgravity = hpg h=ΔPsock/pg 20mmhg = 2666.45 h= 2666.25 / 1050(9.81) = 24.9cm --> 0.249 m --> 0.82ft
what is the flow rate of water that moves at an average speed of 2.8 m/s through a pipe with a diameter of 15cm?
Q=AV Q= π (0.15/2)^2 (2.8) Q=0.0499 m3/s
A pump is used to empty a 6000 L wading pool. The water exits the 2.5-cm-diameter hose at a speed of 2.1 m/s. How long will it take to empty the pool?
Qout= 2.1 m/s ((0.025)^2π/ 4) = 0.00103 m^3/s Qin= ΔV / Δt = 6m^2 / Δt = Qout Δt = 6/0.00103 = 5820.52 sec Δt = 5820.52 sec / 60sec = 97.01 min
A car manufacturer claims that you can drive its new vehicle across a hill with a 47° slope before the vehicle starts to tip. If the vehicle is 2.0 m wide, how high is its center of gravity
Tan47=1/h h= 1/tan47 h= 0.93
How close to the right edge of a 56 kg picnic table can a 70 kg man stand without the table tipping over?
WtDltable = Wmdl 56(0.5) = 70d solve for d D= 56 x g x 0.5 / 70g = 0.4 m 0.55-D = 0.55 - 0.4 = 0.15 m ~ 15 cm
Astronauts in the International Space Station must work out every day to counteract the effects of weightlessness. Researchers have investigated if riding a stationary bicycle while experiencing artificial gravity from a rotating platform gives any additional cardiovascular benefit. What frequency of rotation, in rpm, is required to give an acceleration of 1.4g to an astronaut's feet, if her feet are 1.1 m from the platform's rotational axis?
a= w^2r --> w^2 = ar 1.4(9.81) = w^2 (1.1) solve for w^2 w^2 = 3.53 f= w x 60 / 2π (3.53)(60) / 2π = 33.01 rpm
When you hold your hands at your sides, you may have noticed that the veins sometimes bulge—the height difference between your heart and your hands produces increased pressure in the veins. The same thing happens in the arteries. Estimate the distance that your hands are below your heart. If the average arterial pressure at your heart is a typical 100 mm Hg, what is the average arterial pressure in your hands when they are held at your side?
assume bernoilli principle - wide pipe Phand = Pheart + Pgh Phand= 100 (133Pa) + 1050 (9.81)(1.0) Phand= 23,600.5 Pa
What is the net torque about the axle on the pulley? Where the diameter is 40 cm, force 1= 20 and force 2= 30
diameter =4.0 radius = 4.0 / 2 = 2.0 cm F1= 20 F2 = 30 net torque= τ1 + τ2 τ = F1r - F2r τ = 20(2) - 30(2) τ = -20 ~ -0.2 Nm
water rises in plants through tiny capillaries which have different diameters that range from about 0.01mm to 0.30mm. what is the maximum height to which water can rise in a xylem system?
h= 2𝛿cosθ / pgn h= 2(0.073)(cos(0)) / 1000(9.81)(0.01/2) = 0.00297mm ~ 2.97 m
a glass capillary tube with a diameter of 1.0mm is placed vertically in a dish of mercury. how far is the mercury in the tube depressed below the surface level in the dish?
h= 2𝛿cosθ / pgn h= 2(0.45)(cos(140) / 13600 (9.81)(10^-3/2) = -0.010m
A standard four-drawer filing cabinet is 52 inches high and 15 inches wide. If it is evenly loaded, the center of gravity is at the center of the cabinet. A worker is tilting a filing cabinet to the side to clean under it. To what angle can he tilt the cabinet before it tips over?
height= 52 in width = 15 in TanA= h/w A= arctan(h/w) = arctan (52/15) = 73.19 90=73.19 + x solve for x x= 16.09
Hold your arm outstretched so that it is horizontal. Estimate the mass of your arm and the position of its center of gravity. What is the gravitational torque on your arm in this position, computed around the shoulder joint?
mass of arm= 3kg length of arm = 80 cm --> 0.8m L/2 = 0.8/2 = 0.4 m distance from joint weight acting = mg = 3(9.81) = 29.43 N τ = wd = 29.43(0.4) = 11.77 Nm
In addition to their remarkable top speeds of almost 60 mph, cheetahs have impressive cornering abilities. In one study, the maximum centripetal acceleration of a cheetah was measured to be 18 m/s^2 . What minimum value of the coefficient of static friction between the ground and the cheetah's feet is necessary to provide this acceleration?
mv^2/r = umg --> v^2/r u= 18 m/s^2 / 9.81 = 1.83
A typical horse weighs 5000 N. What fraction of the horse's weight is borne by the front hooves?
nfront + nrear = 5000N nrear = 5000N - nfront @ static equilibrium torque = 0 0=1.8 nfront - (1.1)(5000)
A syringe with a plunger diameter of 2.0cm is attachted to a hydoermic needle with a diameter of 1.5mm. what minimum force must be applied to the plunger to inject a fluid into a vein where the blood pressure is 12 torr
no fluid flow, V=0 Pa-Pb=0 Pa=Pb F/A=12torr F= A(12 torr) F= π (0.02/2)^2 (12 x 133.3) = 0.50N
How easy is it to breathe through a straw? When you breathe deeply, you pull in 4.0 L of air in about 3.0 s. This requires a pressure difference of about 4.0 kPa between the air in your lungs and the outside air. What additional pressure difference is required to pull 20°C air through a straw that is 22 cm long and 0.56 cm in diameter?
radius of straw = d/2 = 0.56/2 = 0.28 cm Q= πΔPr^4 / 8nL = V/t = πΔPr^4 / 8nL ΔP = 8nLV / πR^4t = 8(18.13x10^-6)(0.22m)(4.0L) / π (0.28)^4 (3.0) ΔP = 0.00220 Pa / 1000 = 2.2 x 10^-6
To throw a discus, the thrower holds it with a fully outstretched arm. Starting from rest, he begins to turn with a constant angular acceleration, releasing the discus after making one complete revolution. The diameter of the circle in which the discus moves is about 1.8 m. If the thrower takes 1.0 s to complete one revolution, starting from rest, what will be the speed of the discus at release?
v=wr r= 1.8 m w= 1 revolution / 1.0 sec = 2π/ 1.0 = 6.28 rad/sec v= (6.28 rad/sec)(1.8m) = 11.30 m/sec
The 2.0 kg, uniform, horizontal rod is seen from the side. What is the gravitational torque about the point shown?
w=mg = 2.0(9.81) w= 19.62 ~ 0.196 N left side: 25(0.196)(25/2) = 61.25 N/cm right side: 75(0.196)(75/2) = 551.3 N/cm
a hypodermic needle with an inner diameter of 0.10cm and a length of 3.0 cm is attached to a syringe whose plunger has a cross sectional area of 5.0cm^2. the syringe is filled with salt solution, If a force of 100N is applied to the plunger, what is the flow rate from the needle when the solution is squirted into the air?
ΔP = Ptop - Patm Ptop = 100N/ 5 x 10^-4 = 20 x 10^4 Pa ΔP = 99 x 1000 pa = 32 (1.002 x 10^-3)V(0.03)/(0.001)^2 9900-961V solve for V V=103.017 m/s Q= 103.01 (π(0.001)^2/4) = 8.09 x 10^-5
When you stand quietly, you pivot back and forth a very small amount about your ankles. It's easier to maintain stability if you tip slightly forward, so that your center of gravity is slightly in front of your ankles. The 68 kg man is 1.8 m tall; his center of gravity is 1.0 m above the ground and, during quiet standing, is 5.5 cm in front of his ankles. What is the magnitude of the gravitational torque about his ankles?
τ = 68 (9.81)(5.5/100) = 36.69 Nm
An object's moment of inertia is 2.0 kgm^2. Its angular velocity is increasing at the rate of 4.0 .rad/s What is the net torque on the object?
τ = Iα τ = (2.0)(4.0) = 8.0 Nm
