Presentation on theme: "Warm-up: Centripetal Acceleration Practice"— Presentation transcript:
1Warm-up: Centripetal Acceleration Practice Assume a satellite in low Earth orbit has an acceleration, caused by gravity, of 9.81 m/s2. If the radius of the orbit is approximately the radius of the Earth, 6370 km, find the velocity of the satellite and the time for one orbit.
2Review: Centripetal acceleration A particle undergoing circular motion is accelerating, even if its speed is not changing.The acceleration of an object moving in a circle is toward the center of the circle.It is thus referred to as centripetal acceleration.The magnitude of the centripetal acceleration depends on the velocity of the particle tangent to the circle.Show whirling tennis ball.
6We will begin our study of rotational motion by considering the motion of a rigid body about a fixed axis, or an axis that is moving parallel to itself like a rolling ball.
7Angular PositionBecause the object is rigid, and the axis is fixed, all the points in the object maintain the same relative position.When the object rotates, every point rotates through the same angle.The angle θ, measured relative to some starting position, gives the angular position of every point in the object.
8Angular DisplacementWhen the object rotates on its axis, each point undergoes the same angular displacement Δθ.If we measure the angular displacement in radians, we can easily find the distance traveled by each point, since by definition of radian measurewhere r is the radial distance from the center.
9Angular Displacement Example Two people ride on a carousel. One rides on a horse located 5 meters from the center. The other rides on a swan located 3 meters from the center.When the carousel goes around ¼ of a revolution, how far does each person travel?
10Angular velocityWe define the angular velocity as the rate of change of angular position.Since radians are dimensionless, the dimensions of the angular velocity are T-1.The magnitude of the angular velocity is the angular speed.Counterclockwise rotation corresponds to a positive velocity.
11Angular velocity practice What is the angular speed, in radians/second, of a motor that spinning with rpm?
12Angular accelerationThe angular acceleration is the rate of change of angular velocity.Since radians are dimensionless, the dimensions of angular acceleration velocity are T-2.This acceleration refers to the increase or decrease in rotational speed of the particle.
13Angular and Tangential Quantities Each angular quantity has a corresponding tangential quantity.The arc length s is the distance travelled along the circle, in m .The tangential velocity is the speed of the particle in the direction tangent to the circle, in m/s.The tangential acceleration is the acceleration of the particle tangent to the circle, in m/s2.
14Angular equations of motion for constant acceleration Begin with the definition of angular acceleration.If the acceleration is constant, what is the angular velocity?Then what is the displacement?
15Rotational motion of a rigid body is analogous to linear motion. Straight line motionRotation about a fixed axisLinear position x.Linear displacement DxLinear velocity vLinear acceleration aAngular position q.Angular displacement DqAngular velocity ωAngular acceleration αRemind students not to confuse this analogy with finding the tangential quantities. These are only mention to help them remember the formulas.
16Equations of motion for constant acceleration Straight line motionRotation about a fixed axis
17Problem solving with rotational motion. Convert tangential velocities, speeds etc. to their angular counterparts by dividing by the radius.Solve for the angular motion by using the angular equations of motion just as we did with linear motion.Convert quantities to their tangential counterparts if needed.
18Centripetal acceleration in angular form. We can write the equation for centripetal acceleration in terms of the angular velocity.
19Angular kinematics examples Toast falling off a table usually starts to fall when it makes an angle of 30 degrees with the horizontal, and falls with an initial angular speed of where l is the length of one side. On what side of the bread will the toast land if it falls from a table 0.5 m high? If it falls from a table 1.0 m high? Assume l = 0.10 m, and ignore air resistance.When a turntable rotating at 33 rev/min is turned off, it comes to rest in 26 s. Assuming constant angular acceleration, find the angular acceleration and the angular displacement. If the turntable is 0.20 m in radius, how far would an ant riding on the outside edge have moved in that time?
21Rotational Equilibrium What causes angular acceleration?F1 and F2 are forces equal in size but acting in opposite directions.Will both situations result in angular acceleration?
22Torque A force F acts on mass m located with a rigid body. Only the tangential component affects the rotation of the disk.The farther away from the axis of rotation the force is, the greater the effect on the rotation.
23TorqueThe torque that acts on mass m in the rigid body is the tangential force FT times the distance from the axis of rotation r.Τ=FT r = (F sin φ) r
24TorqueAnother way of looking at the same thing is to say the torque is caused by the entire force, but the distance is only the perpendicular distance of the force to the line of action of the force.This distance, which is r sin φ, is known as the lever arm l.So our formula becomesΤ=F l = F (r sin φ) = Fr sin φ
25Conditions of Equilibrium We learned last year that there are two conditions for equilibrium.The net external force acting on the body must be zero.The net external torque about any point must be zero.
26Torque and Rotational Equilibrium Examples A 700 N window washer is standing on a uniform scaffold supported by a vertical rope at each end. The scaffold weights N and is 3.00 m long. What is the force in each rope when the window washer stands 1.00 m from one end?Does the Sun exert a torque on the Earth relative to the Earth’s rotational axis? Discuss.
27Rotational equilibrium example A 700 N window washer is standing on a uniform scaffold supported by a vertical rope at each end. The scaffold weights N and is 3.00 m long. What is the force in each rope when the window washer stands 1.00 m from one end?Answers: 333 N and 567 N
28Rotational Dynamics: Newton’s 2nd Law for Rotation Have student sit in chair holding masses. Demonstrate how different amounts of torque are needed to cause a certain rotational acceleration. Or do soup demo.
29Newton’s Second Law for Rotational Motion Consider a mass m in a rigid body that is free to rotate.The mass is acted on by a force F.What does the radial component do?What does the tangential force do?The mass will undergo a tangential acceleration at.This acceleration is in addition to the constant centripetal acceleration.
33Newton’s Second Law for Rotational Motion We can add up the torques on every mass mi in the object to derive an equation for the entire rigid object.Since this is a rigid object, α is the same for every particle. Thus we can remove it from the sum.
34Newton’s Second Law for Rotational Motion The internal torques add up to zero, so the left hand side of the above is just the net external torque.The quantity in brackets above depends on the mass and radius of each particle, both of which are fixed for a rigid object and choice of axis. So, that sum is also fixed for a given object and rotational axis. It is called the rotational inertia.
35Newton’s Second Law for Rotational Motion Where I = rotational inertiaRotational inertia is also called the moment of inertia.
40Parallel Axis Theorem The parallel-axis theorem states Where ICM is the rotational inertia about an axis containing the Center of MassI is the rotational inertia about an axis parallel to ICMM = total mass and h = distance between axes.
47Rotational Kinetic Energy We can apply the formula for kinetic energy to rotation as well. In a rotating rigid body, each particle moves with a velocity vt=rω.
48Rotational Kinetic Energy Example Calculate the rotational kinetic energy of the Earth, and compare that to the kinetic energy of the Earth’s motion around the Sun. Assume the Earth is a sphere of radius 6.4 x 106 m and mass 6.0 x 1024 kg. The radius of the Earth’s orbit is 1.5 x 1011 m.
50Rolling ObjectsConsider an object with a circular cross section that rolls without slipping across a flat surface.The point of contact between the wheel and the ground moves a distance s = Rφ.Since the center of mass remains above the point of contact, it also moves s=Rφ.
52Rolling ObjectsFor an object that rolls without slipping, the motion of the center of mass is described by the same equations as an object rotating about a fixed axis.Newton’s Second Law for rotation also holds, if the torques are calculated about an axis containing the center of mass.τNET, CM = ICMα
53Rolling ObjectsThe total kinetic energy of a moving object is the kinetic energy of the center of mass plus the kinetic energy of the particles relative to the CM.Here, that means the rotation kinetic energy plus the kinetic energy of the CM motion.
54A bowling ball of radius 11 cm rolls without slipping up a ramp A bowling ball of radius 11 cm rolls without slipping up a ramp. If the initial speed of the ball is 2 m/s find the height h that the ball reaches when it momentarily comes to rest.
55A cue stick hits a cue ball horizontally a distance d above the center of the ball. Find the value of d for which the cue ball will roll without slipping from the beginning. Express your answer in terms of the radius R of the ball.