UCONN Physics 1 & 2 (Scanlon) Chapter 7 Notes: Rotational Motion
Describe the following symbol: α - name - used for - units - variable counterpart (kinematics)
- alpha - angular acceleration - radians/second² (π/s²) - a (acceleration)
Describe the following symbol: ω - name - used for - units - variable counterpart (kinematics)
- omega - angular velocity - radians/second (π/s) - v (velocity)
A board is supported by a wooden sawhorse on one end and held up by a hand on the other. The mass of the board is 20 kg and the length is 3 m. Answer the following: - Where is the center of mass? - Where is the axis of rotation? - What happens if you release the end supported by your hand? - What is the torque if you release the end supported by your hand?
- the middle of the board, 1.5 m from either side - the edge on the sawhorse - the board rotates as it falls - τ = r*F*sinθ F = mg = 30 kg * 9.8 m/s^2 = 196 N τ = 1.5 m * 196 N * sin 90° τ = 294 mN
Describe the following symbol: θ - name - used for - units - variable counterpart (kinematics)
- theta - angular displacement - radians (π) - x (position)
Describe the following equations in terms of the variables for angular motion: - 3 - 6 - 7
- ωf = ωi + αt - θf = θi + ωit + αt² - ωf² = ωi² + 2α(∆θ)
What is always the angle if the object is being pushed not pulled?
90°
A bottle cap is rotated with velocity v as a force is exerted on it tangentially to the radius. What is the angle and when is it the maximum?
90° angle, maximum torque
Describe the following for a 20 kg board supported on one end by a sawhorse and on the other by a 10 kg hand where everything is measured from a pivot point, a = 0, and εF = 0: Fg, F(hand), ετ
F(hand) = 10 kg * 9.8 m/s^2 = 98 N Fg = mg = 9.8 m/s^2 * 20 kg = 198 N no τ at pivot point; τ = 0 so equilibrium ετ= 0 = τ(Fg) + τ(hand) 0 = - 294 mN + 294 mN
What is the formula for moment of inertia?
I = mr²
A string is wrapped around a uniform solid cylinder of radius r. The cylinder can rotate freely about its axis. The loose end of the string is attached to a block. The block and cylinder each have mass m. Find the magnitude α of the angular acceleration of the cylinder as the block descends.
I(pulley) = 1/2mr², a = rα, ετ = Iα, εFy = mg - T rFsinθ = rT = 1/2mr²α r(mg - mrα) = 1/2mr²α mgr - mr²α = 1/2mr²α mgr = 1/2mr² + mr²α = 3/2mr²α m and r cancel; g = 3/2rα α = (2g/3)/r
A bicycle with 0.80-mm-diameter tires is coasting on a level road at 5.6 m/s . A small blue dot has been painted on the tread of the rear tire. a) What is the angular speed of the tires? b) What is the speed of the blue dot when it is 0.8 m above the road? c) What is the speed of the blue dot when it is 0.40 mm above the road?
a) rω = v so ω = 5.6 m/s/(0.4 m), where 0.4 m is the diameter; ω = 14 m/s b) 2v so 11 m/s ***at top = 2v, at bottom v = 0, on sides vector combo so square root of 5.6² + 5.6² c) vector combo; 7.9 m/s
A 1.70-mm-long barbell has a 25.0 kg weight on its left end and a 32.0 kg weight on its right end. a) If you ignore the weight of the bar itself, how far from the left end of the barbell is the center of gravity? b) Where is the center of gravity if the 9.00 kg mass of the barbell itself is taken into account?
a) use x(cm) = m1x1 + m2x2/m(T) (25 kg)(0) + (32 kg)(1.7 m)/(57 kg) xc = 0.954 m b) x(cm) = m1x1 + m2x2 + m3x(r)/m(T) 25(0) + 32(1.7) + 9*(0.85)/(66 kg)
What should you do if more objects/masses are added?
add moments of inertia
Part complete If Luis pulls straight down on the end of a wrench that is tilted θ = 30° above the horizontal and is r = 29 cm long, what force must he apply to exert a torque of -13 N⋅m?
angle between force and straight trajectory of wrench; 180-30 = 120° τ = rFsinθ -13 = .29 m * F * sin 120
What is the official term for an object returning to its original position via angular motion?
angular displacement
What two directions can torque, as a vector, move in?
counterclockwise (CCW), negative clockwise (CW), positive
Is it easier to exert a force on a doorknob or a door hinge?
doorknob (farther away/longer radius)
What should you do if the force is exerted at an angle?
find the vertical component of the angle (the horizontal component has a force of 0 so no torque)
torque
how well a force causes a rotation
When is the only time you should use another formula for moment of inertia?
if provided for a specific object on a test
What happens to force if you exert a force 2x the distance away from the axis of rotation?
it requires half the force
A door on a hinge is perpendicular to the wall. You grasp it at 90° and push the door. What is the force for the torque?
sin 90° = 1 (max value) so the equation is rF
Where would you want to balance a board?
the center of mass
What must happen in order for an object to be in equilibrium under the action of its own weight?
the center of mass of the object must lie directly above the base of the object so that torque due to weight will be 0
Where is velocity fastest on a circle?
the outside of the circle; slowest on the inside
A 30-kg child stands at one end of a floating 20-kg canoe that is 5.0-m long and initially at rest in the water. The child then slowly walks to the other end of the canoe. How far does the canoe move in the water, assuming water friction is negligible?
the position of the center of mass does not change center of mass of canoe = 2.5 m walking on canoe = internal force so as the boy moves one way the canoe must move the other way if he is at x = 0 initially then xcm = (0)(30) + (2.5)(20)/50 = 1 m he must walk so he is 1 m from the other side to balance so 5-(1+1) = 3m
What happens if torque is cancelled out and τ = 0
there is no rotation
What takes the place of force in angular motion?
torque
Given a plank half over the side of a ledge with a weight on either side, the given values of the radii and a given mass, and. one unknown mass, what is the mass m needed to keep the board from tipping over?
torque due to mass = τ = r*F*sin θ since both are at 90° angles the equation is r1F1 = r2F2
When does torque increase?
when it is exerted farther away from the axis of rotation
If a tire iron is attached to the side of a tire, when will it have maximum torque?
when pushed up or down
What is the formula for the location of the center of mass?
x(cm) = m1x1 + m2x2/m(T)
A 4.9-mm-long, 550 kg steel beam extends horizontally from the point where it has been bolted to the framework of a new building under construction. A 71.0 kg construction worker stands at the far end of the beam. What is the magnitude of the torque about the point where the beam is bolted into place?
ετ = τ(beam) + τ(person) ετ = (2.45 m)(550 kg)(9.8 m/s) + (4.9 m)(71 kg)(9.8 m/s)
What is another equation for θ?
θ = s/r
You are looking down 100 kg merry-go-round with a radius of 3m. The merry-go-round is being pushed tangentially by a force of 25 N. What is the acceleration of the merry-go-round? The equation for I is 1/2mr²
τ = (3 m)(25 N)(sin 90°) = 75 mN CCW α = τ/I I = 1/2mr² = (1/2)(100 kg)(3)² = 450 kg*m² α = 75 mN/450 kg*m² α = 0.167 π/s²
You are looking down 100 kg merry-go-round with a radius of 3m. The merry-go-round is being pushed tangentially by a force of 25 N and a 50 kg kid sits on the equipment 2 m from the radius. What is the acceleration of the merry-go-round? The equation for I is 1/2mr²
τ = 75 mN CCW I = I(disk) + I(kid I = 450 kg*m² + mr² (50 kg)(2)² = 200 kg*m² α = 75 mN/650 kg*m² α = 0.115 π/s²
A uniform rod is 2.0 m long. It is hinged to a wall at its left end, and held in a horizontal position at its right end by a vertical very light string, as shown in the figure. What is the angular acceleration of the rod at the moment after the string is released if there is no friction in the hinge?
τ = Ia; I = 1/3ml² 1/3ml²α = rg α = 3rg/l² = (3)(1)(9.8)/(2²)
Translate F = ma to angular motion
τ = Iα τ = torque I = moment of inertia α = angular acceleration
A cinder block of mass m = 4.0 kg is hung from a nylon string that is wrapped around a frictionless pulley having the shape of a cylindrical shell, as shown in the figure. If the cinder block accelerates downward at 4.90 m/s2 when it is released, what is the mass M of the pulley?
τ = Iα = rT; rT = 1/2mr²α T = 1/2mr²α/r = 1/2mrα; T = 1/2 ma Fg - T = ma; 1/2ma = Fg - ma 1/2m(p) (4.9) = 4(9.8) - 4(4.9) m(p)(2.45) = (4 kg)(4.9 m/s²) m(p) = 8 kg
What is the equation for torque?
τ = r * F * sin θ
A 3m board of 30 kg is attached to a sawhorse on one end and lifted a few degrees above the vertical by a 10 kg hand (pushed upward at 90°). What is τ(hand)?
τ(hand) = r*F*sinθ εF = F(board) + F(hand) = 294 N + 98 N F(hand) = 98 N (10 kg * 9.8 m/s^2) τ = 294 N = (3 m)*F(hand)*sin 90°
A grinding wheel at rest has an acceleration of 2 radians/s² and t = 3 s. What is ωf and θ?
ω = ωi + αt θ rad/s = 0 rad/s + 2 rad/s²(3s) ωf = 6 rad/s θf = θi + ωit + αt² θf = 0 + 0 + 1/2(2 rad/s²)(3 s)² θ = 9 radians (in 3 s)
A computer hard disk starts from rest, then speeds up with an angular acceleration of 190 rad/s2 until it reaches its final angular speed of 7200 rpm. How many revolutions has the disk made 10.0 ss after it starts up?
ω(f) = 7200 rpm * 2π/1 rev * 1 m/60 s = 754 rad/s 3) 754 rad/s = 0 + 190 rad/s²(t); t = 3.96 s 6) θ = 0 + 0 + 1/2(190)(3.96)² = 1489.8 rad 10 - 3.96 = 6.04s; 1) = ∆θ = ωt = (754 rad/s)(6.04 s) = 4554.2 rad 1489.8 + 4554.2 = 6044 rad; 6044 rad * 1 rev/2π = 962 turns
What is one radian approximately equal to?
≈ 60°
What is the value in radians if an object makes two revolutions and returns to its original position? What is the distance in meters for one revolution if the circumference is 2 meters?
4π radians; x = 2m(2π radians) = 4π m
Describe the formulas for the following: . a(t) . a(r)
. a(t) = rα . a(r) = vt²/r
Find the following from their arc measurements: . x . s . v . a
. x = rθ . s = rθ . v = rω . a = rα
If the radius of a sector is 0.4 m , find the following: . x when θ is 9 radians . v when ω is 6 radians/s
. x = rθ = (0.4)(9 radians) = 3.6 m . v = rω = (0.4)(6 radians/s) = 2.4 m/s
A door on a hinge is perpendicular to the wall. You grasp it at 180° and push, then pull straight away from the wall. What is the torque? What happens to the torque if you push a door at 90°?
0 because the force exerted is 0
What is the radius if you lift up the board with your finger on one end? Describe the force.
1.5 m; the radius Less force and same/greater impact
What is one revolution equal to?
2π
What is the value in radians if an object returns to its original position?
2π
What is the formula for circumference?
2πr
How do you balance an object to find the center of mass?
mx - mx, in which x is the radius
A bucket hangs from a pulley and has a rope draped over it. What is the tension of the bucket? What is the acceleration?
pulley rotating so it doesn't fall at 9.8 m/s^2 a) τ = Iα; τ = rFsinθ; I = 1/2mr² r*F*sin θ = 1/2mr² T = 1/2mrα b) εFy = m(b)a - T = m(b)a T = 1/2m(p)rα so m(b)a - 1/2m(p)rα = m(b)a a = rα so α = a/r m(b)a - 1/2m(p)a = m(b)a a = [m(b)g]/[1/2 m(p) + m(b)]
What does circular motion do?
return to its original position = angular motion
