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Circular Motion 5.2 Uniform Circular motion 5.3 Dynamic of Uniform Circular Motion HW4: Chapt.5: Pb.23, Pb.24, Pb.30, Pb.33, Pb.36, Pb.53- Due FRIDAY, OCT. 2 Applications of Newton’s Laws Involving Friction Example 5-3: Pulling against friction. A 10.0-kg box is pulled along a horizontal surface by a force of 40.0 N applied at a 30.0° angle above horizontal. The coefficient of kinetic friction is 0.30. Calculate the acceleration. Uniform Circular Motion—Kinematics Uniform circular motion: motion in a circle of constant radius at constant speed Instantaneous velocity is always tangent to the circle. Linear vs. Circular motion: Linear motion Relationship Angular motion Position Angle Displacement Angular displacement Average velocity Average angular velocity Instantaneous velocity Instantaneous angular velocity Uniform circular motion Uniform circular motion means Constant rotation speed • How far? – Angular displacement Dq = qf - qi – The size of the angle swept out by the motion – Typically “+” indicates counterclockwise – Units - radians Go around once: 2π radians 360 degrees 1 revolution r y f i x Uniform circular motion How fast? – Angular velocity y – This is constant for Uniform circular motion – Units: rad/sec rpm-revolutions per minute rad/s r f i x Relationship between angular and linear motion • How far does it go? – Angular displacement, to linear motion, s. s = rDq – Here r is the radius of the circle in meters, and s is the distance traveled in meters (or arc length). is the angular displacement in radians s r = Dq since s/r is unitless, radians are not a physical unit, and do not need to balance like most units. s r y f i x Relationship between angular and linear motion How fast does it go? – Angular velocity , to linear velocity, v v avg = Ds Dt = rDq Dt = rw avg – Direction of v is tangent to the circle – Units : v v m/sec must be in rad/s y r v f i x • A) Objects 1 and 2 have the same linear velocity, v, and the same angular velocity, . • B) Objects 1 and 2 have the same linear velocity, v, and the different angular velocities, . • C) Objects 1 and 2 have different linear velocities, v, and the same angular velocity, . • D) Objects 1 and 2 have different linear velocities, v, and the different angular velocities, . Question • Two objects are sitting on a horizontal table that is undergoing uniform circular motion. Assuming the objects don’t slip, which of the following statements is true? 1 2 Question • Two objects are sitting on a horizontal table that is undergoing uniform circular motion. Assuming the objects don’t slip, which of the following statements is true? • A) Objects 1 and 2 have the same linear velocity, v. • B) Object 1 has a faster linear velocity than object 2. • C) Object 1 has a slower linear velocity than object 2. 1 2 Period and frequency w = 2pf 1 T= f Linear vs. Circular motion: Linear motion x Position Displacement Average velocity Instantaneous velocity Angular motion Dx = x f - xi Dx Dt Dx = lim Dt®0 Dt vavg = vinst Angle Angular displacement Average angular velocity Instantaneous angular velocity q Dq = q f - qi Dq wavg = Dt Dq winst = lim Dt®0 Dt Dynamics of Uniform Circular Motion Velocity can be constant in magnitude, and we still have acceleration because the direction changes. • Direction: towards the center of the circle Newton’s second law • Whenever we have circular motion, we have acceleration directed towards the center of the motion. • Whenever we have circular motion, there must be a force towards the center of the circle causing the circular motion. åF v = ma 2 r r ar = rw = 2 r Dynamics of Uniform Circular Motion There is no centrifugal force pointing outward; what happens is that the natural tendency of the object to move in a straight line must be overcome. If the centripetal force vanishes, the object flies off at a tangent to the circle. Dynamics of Uniform Circular Motion Example 5-11: Force on revolving ball (horizontal). Estimate the force a person must exert on a string attached to a 0.150-kg ball to make the ball revolve in a horizontal circle of radius 0.600 m. The ball makes 2.00 revolutions per second. Ignore the string’s mass.