Review questions. Before the collision, 70 kg ball is stationary. Afterward, the 30 kg ball is stationary and 70 kg ball is moving to the right.

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1 Review questions Before the collision, 70 kg ball is stationary. Afterward, the 30 kg ball is stationary and 70 kg ball is moving to the right. 30 kg 70 kg v (a) Is this collision elastic? (b) Find the final speed of the 70 kg ball.

2 Ch. 8: Impulse, momentum, collisions Momentum, p = mv (inertia of moving object) 2 Impulse: J = F t or Fdt 1 gives momentum change: J = p2 p1 Momentum Conservation: total P = p + p +... conserved for closed systems. Elastic vs. inelastic collisions: K conserved for elastic m1r 1 + m2r Center of mass, rcm = m1 + m CM velocity: CM moves as particle with momentum P = Mv cm (also CM has important role for acceleration of solid rotors) A B

3 Exam 3 formula sheet (2 nd page is moments of inertia)

4 Before the collision, 70 kg ball is stationary. Afterward, the 30 kg ball is moving vertically with speed v/2. 30 kg 70 kg v (a) Is this collision elastic? (b) Find the final speed and direction of the 70 kg ball.

5 A thin-walled hollow sphere, m = 12 kg & diameter = 48.0 cm, is rotating according to, θ = At 2 + Bt 4, with A = 1.50 and B = 1.00, t given in seconds. (a) What are the units of A and B? (b) At t = 2.00 s, find the angular momentum, and the torque on the sphere.

6 Chapter 9: Rotations Angular velocity, acceleration: ω = dθ angular velocity, α = dω angular acceleration. dt dt 2 Linear vs. angular relations, Vectors: v = rω atan = rα arad = ω r ω, α along axis in right hand rule direction. Constant angular acceleration: familiar formulas. θ = θ 0 + ω 0 t αt 2 analog of similar for other constant-a relations. Energy and Moment of inertia: x = x 0 + v 0 x t a x t 2 K = 1 2 Iω 2 analog of K = 1 2 mv2 I 1 = 2 mr 2 moment of inertia.

7 I = mr 2 moments of inertia same as for KE.

8 A solid spool, mass M, radius R, is held steady by a string wound around its perimeter. Spool rests with no slipping on 45 ramp. Find T in the string and the friction force. Other questions: (a) Suppose hollow cylinder instead of solid spool, but same mass M. Is T same or different? (b) Suppose string pulled s.t. string has constant velocity v, and spool rolls uphill with no slipping. Find T in this case.

9 Rotational Mechanics & Equilibrium KE = 1 2 Iω 2 (energy) τ = Iα W = τθ L = Iω (torque) Rotational form of F = ma (work) W = Fs (momentum) p = mv Substitute: θ for x I for m τ for F ω for v α for a L for p Equilibrium: F x = 0 F y = 0 τ = 0 (any axis) same as CM x = w x w = m x m cg i i i i i i

10 A solid spool, mass M, radius R, held steady by a string wound around its perimeter. Spool rests with no slipping on 45 ramp. String is cut allowing ball to roll downward without slipping. (a) Find acceleration down the ramp. (b) Find the friction force.

11 A mass is hanging from the end of a horizontal bar which pivots about an axis through it center, but it being held stationary. The bar is released and begins to rotate. As the bar rotates from horizontal to vertical, the magnitude of the angular acceleration α of the bar.. A) increases B) decreases C) remains constant

12 A mass is hanging from the end of a horizontal bar which pivots about an axis through it center, but it being held stationary. The bar is released and begins to rotate. As the bar rotates from horizontal to vertical, the magnitude of the torque on the bar.. A) increases B) decreases C) remains constant

13 Ball has m = 3.0 kg and initial v = 2.0 m/s. The rotor has total m = 5.0 kg, consists of a 3.0 kg rod, length L = 1.0 m, with a small 2.0 kg mass attached at the end. The ball strikes the rotor at its midpoint & sticks to it. pivot v (a) Is energy conserved in the collision? (b) What is the angular velocity of the rotor just after the collision? (c) What maximum angle does rotor attain?

14 Exam 3 formula sheet (2 nd page is moments of inertia)

15 Find the center of mass. 30 m 50 kg 40 m 20 kg 30 kg

16 Find the center of mass velocity, if the lower two masses are stationary. 30 m 50 kg 15 m/s 40 m 20 kg 30 kg

17 Consider a sudden elastic collision between the ball and the block. What is the maximum angle of rise of the ball when it rebounds after collision? (a) same as original θ. (b) larger. (c) smaller.

18 A grinding wheel is spinning freely at 30.0 rpm. Then, an iron bar is pushed against its edge at constant force, causing it to stop in 4.5 s. (a) Is the angular acceleration constant? (b) Find the total angle through which the wheel turns while stopping.

19 Consider masses arranged as below, with M > m. The pulley has mass m p, and is a solid disk, radius r. No friction a) Compute the acceleration and tensions T 1 and T 2 on opposite sides of the pulley.

20 Consider masses arranged as below, with M > m. The pulley has mass m p, and is a solid disk, radius r. No friction a) Compute the acceleration and tensions T 1 and T 2 on opposite sides of the pulley.

21 Consider masses arranged as below, with M > m. The pulley has mass m p, and is a solid disk, radius r. No friction a) Compute the acceleration and tensions T 1 and T 2 on opposite sides of the pulley.

22 Consider masses arranged as below, with M > m. The pulley has mass m p, and is a solid disk, radius r. No friction a) Compute the acceleration and tensions T 1 and T 2 on opposite sides of the pulley. b) Using energy methods, find the velocity after M has fallen a distance h.

23 Consider masses arranged as below, with M > m. The pulley has mass m p, and is a solid disk, radius r. No friction a) Compute the acceleration and tensions T 1 and T 2 on opposite sides of the pulley. b) Using energy methods, find the velocity after M has fallen a distance h.

24 Consider masses arranged as below, with M > m. The pulley has mass m p, and is a solid disk, radius r. No friction a) Compute the acceleration and tensions T 1 and T 2 on opposite sides of the pulley. b) Using energy methods, find the velocity after M has fallen a distance h. Note acceleration methods also work to find solution

25 The non-uniform bar has mass 30 kg and its CM is 0.75 m from the left end. The bar length is 2.00 m. Cables are attached to left end and 0.4 m from right end. Find the cable tensions.

26

27 The ideal pulley is massless, and attached to a fixed axle. The rope supports a mass M 2 as shown. The other end of the rope is tied to the plank, making an angle of 30. The plank is also supported by a fulcrum at a distance L/4 from the left end of the plank, as shown. In terms of M 1, find the value of M 2 that will allow equilibrium. M 2 M 1

28 The ideal pulley is massless, and attached to a fixed axle. The rope supports a mass M 2 as shown. The other end of the rope is tied to the plank, making an angle of 30. The plank is also supported by a fulcrum at a distance L/4 from the left end of the plank, as shown. In terms of M 1, find the value of M 2 that will allow equilibrium. Also find force on fulcrum. M 2 M 1

29 If box is 3m high and 2m wide, find initial angular acceleration. Assume the box does not slip on the support point. Also find the instantaneous force components exerted by the fulcrum. M

30 I = mr 2 moments of inertia same as for KE.

31 A tire (illustrated) has fallen from a cart and is rolling on a horizontal road with constant center of mass velocity v. Find the instantaneous velocity vector and acceleration vector for the part of the tread at the edge of the tire (arrow) at the front edge even with the center of the tire. M

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Chapter 8 Lecture Notes Chapter 8 Lecture Notes Physics 2414 - Strauss Formulas: v = l / t = r θ / t = rω a T = v / t = r ω / t =rα a C = v 2 /r = ω 2 r ω = ω 0 + αt θ = ω 0 t +(1/2)αt 2 θ = (1/2)(ω 0 +ω)t ω 2 = ω 0 2 +2αθ τ

= o + t = ot + ½ t 2 = o + 2 Chapters 8-9 Rotational Kinematics and Dynamics Rotational motion Rotational motion refers to the motion of an object or system that spins about an axis. The axis of rotation is the line about which the

PHYSICS 221 Fall 2016 EXAM 2: November 02, :15pm 10:15pm. Name (printed): Recitation Instructor: Section #: PHYSICS 221 Fall 2016 EXAM 2: November 02, 2016 8:15pm 10:15pm Name (printed): Recitation Instructor: Section #: INSTRUCTIONS: This exam contains 25 multiple-choice questions, plus 2 extra-credit questions,

16.07 Dynamics. Problem Set 10 NAME :..................... Massachusetts Institute of Technology 16.07 Dynamics Problem Set 10 Out date: Nov. 7, 2007 Due date: Nov. 14, 2007 Problem 1 Problem 2 Problem 3 Problem 4 Study Time Time Spent AP Physics 1- Torque, Rotational Inertia, and Angular Momentum Practice Problems FACT: The center of mass of a system of objects obeys Newton s second law- F = Ma cm. Usually the location of the center

Physics for Scientist and Engineers third edition Rotational Motion About a Fixed Axis Problems A particular bird s eye can just distinguish objects that subtend an angle no smaller than about 3 E -4 rad, A) How many degrees is this B) How small an object can the bird just distinguish when flying

Rotation and Translation Challenge Problems Problem 1: Rotation and Translation Challenge Problems Problem 1: A drum A of mass m and radius R is suspended from a drum B also of mass m and radius R, which is free to rotate about its axis. The suspension is

Rotational Dynamics Smart Pulley Rotational Dynamics Smart Pulley The motion of the flywheel of a steam engine, an airplane propeller, and any rotating wheel are examples of a very important type of motion called rotational motion. If

Phys 2210 S18 Practice Exam 3: Ch 8 10 1. As a 1.0-kg object moves from point A to point B, it is acted upon by a single conservative force which does 40 J of work during this motion. At point A the speed of the particle is 6.0 m/s and the 1. rpm is about rad/s. 7. ROTATIONAL MOTION 2. A wheel rotates with constant angular acceleration of π rad/s 2. During the time interval from t 1 to t 2, its angular displacement is π rad. At time t 2 第 1 頁, 共 7 頁 Chap10 1. Test Bank, Question 3 One revolution per minute is about: 0.0524 rad/s 0.105 rad/s 0.95 rad/s 1.57 rad/s 6.28 rad/s 2. *Chapter 10, Problem 8 The angular acceleration of a wheel