#$&* course phy 201 Aug 1 4:20pm If your solution to stated problem does not match the given solution, you should self-critique per instructions at http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.
.............................................
Given Solution: `aGOOD STUDENT ANSWER: A point moving around a circle can be represented by two perpendicular lines whose intersection is that point of constant velocity. The vertical line then is one that moves back and forth, which can be synchronized to the oscillation of the pendulum. The figure below depicts the reference circle. The vector from the origin to the circle is called the radial vector. The tip of this vector is on the circle, as is called the reference point. A vertical line through the reference point extends down to the x axis, and a horizontal line through the reference point extends over to the y axis. The points where these lines meet the axes are the x and y coordinates of the reference point.  As the reference point and the radial vector move around the circle, the x and y coordinates change. If the reference point is moving counterclockwise around the circle, then starting from the position shown in the figure, the x coordinate will decrease. We imagine an object which moves in such a way that its x coordinate coincides with that of the reference point; in this sense the object 'tracks' the reference point as the object moves along the x axis. • When the reference point reaches the y axis its x coordinate will be zero. • When the reference point reaches the negative x axis the x coordinate will be at its extreme value, i.e., a point as far from the origin as possible for a point moving around this circle. • As the reference point continues moving around the circle an object tracking its x coordinate begins to move to the right, slowly at first them more and more quickly as it approaches the origin. • The x coordinate is again 0 when the reference point reaches the negative y axis. At that point the object has its maximum speed. • As the reference point approaches the positive x axis the x coordinate changes more and more slowly, reaching an extreme point when the reference point reaches the positive x axis. Thus as the reference point continues to move around the circle, the object will continue to oscillate back and forth along the x axis. An object which moves in this manner is said to be undergoing simple harmonic motion. A simple pendulum, oscillating freely with an amplitude much less than its length, is modeled in this manner by a reference circle whose radius is equal to its amplitude of motion. • The time required for one complete trip around the reference circle corresponds to the period of the pendulum (i.e., to the time required for one complete oscillation of the pendulum). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: At what point(s) in the motion a pendulum is(are) its velocity 0? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The two ends, when coming to a brief stop, and then falling back. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: GOOD STUDENT ANSWER: The pendulum has two points of v= 0. One at each end as it briefly comes to a stop to begin swinging in the opposite direction. PARTIALLY CORRECT STUDENT ANSWER: equilibrium, and at each extreme points of the oscillation, because the pendulum briefly stops at these points INSTRUCTOR COMMENT: If a pendulum is not swinging, then its velocity at the equilibrium position is zero. However if, as in the present case, it is swinging, then its passes thru the equilibrium position with a nonzero velocity. Furthermore its velocity as it passes through equilibrium is the maximum velocity it attains during its motion. ********************************************* Question: At what point(s) in the motionof a pendulum is(are) its speed a maximum? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The mid-point confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The pendulum swings from one extreme point to the other, then back, repeating this motion again and again. Halfway between the extreme points is the equilibrium point, the point at which the pendulum would naturally hang if it wasn't moving. It is at this halfway point that the pendulum reaches its maximum velocity. • As we will see, it is at the equilibrium position, and only at the equilibrium position, where the velocity of the pendulum is equal to the velocity of the point on the reference circle. • This could be reasoned out from energy considerations. The equilibrium position is the position at which the pendulum's mass is lowest, and so it the point at which its gravitational potential energy is lowest. Assuming that no nonconservative forces act on the pendulum, this is the point at which its kinetic energy is therefore highest. GOOD STUDENT ANSWER: The mid point velocity of the pendulum represents its greatest speed since it begins at a point of zero and accelerates by gravity downward to equilibrium, where it then works against gravity to finish the oscillation. GOOD STUDENT DESCRIPTION OF THE FEELING: At the top of flight, the pendulum 'stops' then starts back the other way. I remember that I used to love swinging at the park, and those large, long swings gave me such a wonderful feeling at those points where I seemed to stop mid-air and pause a fraction of a moment. Then there was that glorious fall back to earth. Too bad it makes me sick now. That was how I used to forget all my troubles--go for a swing. *&*& INSTRUCTOR COMMENT: That extreme point is the point of maximum acceleration. ** STUDENT QUESTION I thought that the equilibrium point was where the velocity was at zero??? Is this also where the pendulum reaches its maximum velocity??? I figured that it would be gaining the most speed as it went from a zero velocity and was released- maybe when it leaves the amplitude point? INSTRUCTOR RESPONSE The equilibrium point is the point at which the pendulum naturally hangs while stationary. When it's moving, the equilibrium point is still the point at which the pendulum would naturally hang if it was stationary. For a moving pendulum that point coincides with the 'low point' of its arc. I refer to the points where velocity is for an instant zero as the 'extreme points' of its motion. At these points its distance from the equilibrium position is equal to its amplitude. The pendulum does speed up most quickly when it begins its swing back from its equilibrium point. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `qHow does the maximum speed of the pendulum compare with the speed of the point on the reference circle? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Equilibrium, they’re equal. Other points, the pendulum is slower. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** At the equilibrium point the pendulum is moving in the same direction and with the same speed as the point on the reference circle. At any other point it is moving more slowly. University Physics Note: You can find the average speed by integrating the speed function, which is the absolute value of the velocity function, over a period and then dividing by the period (recall from calculus that the average value of a function over an interval is the integral divided by the length of the interval). You find rms speed by finding the average value of the squared velocity and taking the square root (this is the meaning of rms, or root-mean-square). ** STUDENT QUESTION: what about at any point other than equilibrium, will the speed hold the same as for a circle where speed closest to the reference point is slower than at the radius INSTRUCTOR RESPONSE: The point on the reference circle has constant speed. Unless the reference point is moving in the same direction as the pendulum, the part of its speed in the direction of the pendulum's motion is less than its speed on the reference circle. Thus, at only two points on the reference circle is the speed of the pendulum as great at the speed of the reference point. Those points coincide with the equilibrium position of the pendulum (i.e., at the equilibrium position the motion of the reference-circle point is in the same direction as that of the pendulum). The figure below depicts the reference circle along with the radial vector r , the vector from the center of the circle to the reference point. This particular vector corresponds to an angular position of about 35 degrees.  The next figure depicts as well the velocity vector v for the reference point, assuming that the reference point is moving in the counterclockwise direction. The velocity of the reference point is tangent to the circle, and hence perpendicular to the radial vector.  The final figure adds the centripetal acceleration vector a , which points back along the radial vector toward the origin. In the figure the acceleration vector is slightly offset from the radial vector, to make it clearly visible. In reality this vector would lie right over the radial vector.  If this figure models the motion of an object oscillating back and forth along the x axis, then the velocity of the object corresponds to the x component of v. It is clear that in this picture the x component of v is of lesser magnitude than v. That is, the object is moving more slowly than the reference point. If the radial vector was horizontal, then the velocity v of the reference point would be vertical and its x component would be zero. The object would at that instant be stationary. If the radial vector was vertical, then the velocity v of the reference point would be horizontal, and its x component would be of the same magnitude as v. The object would be moving at the same speed as the reference point and this would be its maximum speed. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `qHow can we determine the centripetal acceleration of the point on the reference circle? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (angular velocity*radius)^2/r confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** Centripetal acceleration is v^2 / r. Find the velocity of a point on the reference circle (velocity = angular velocity * radius). The acceleration of the actual pendulum (which we assume moves along the x axis) is the x component of the centripetal acceleration of the reference point. • When the pendulum is at one of its extreme positions, the reference circle is on the x axis and the centripetal acceleration vector points back along the x axis toward the origin, so at this point the acceleration of the pendulum is equal to the centripetal acceleration of the reference point. • When the pendulum is at its equilibrium position the reference point lies directly above or below the pendulum, with its centripetal acceleration vector pointing in the vertical direction. The centripetal acceleration vector therefore has no component in the direction of the pendulum's motion, and the pendulum is for that instant not accelerating. • At any other position the centripetal force vector has a nonzero component in the x direction which has a magnitude less than that of the vector. This component is identical to the acceleration of the pendulum.** STUDENT QUESTION If we want to find the centripetal acceleration of a circle we can use the formula a_cent= v^2/rThe only thing I wonder about is if we would know what the radius of the circle is or is there a formula we could use to solve for this? INSTRUCTOR RESPONSE The radius of the reference circle is equal to the amplitude of motion. When a pendulum moving along the x axis it at its maximum position x = A, the reference point is passing through the positive x axis, and the r vector is directed along the x axis. Thus the magnitude of the r vector is equal to the amplitude A, and the radius of the circle is therefore equal to the amplitude. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `qHow can we determine the centripetal acceleration of the point on the reference circle? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (angular velocity*radius)^2/r confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** Centripetal acceleration is v^2 / r. Find the velocity of a point on the reference circle (velocity = angular velocity * radius). The acceleration of the actual pendulum (which we assume moves along the x axis) is the x component of the centripetal acceleration of the reference point. • When the pendulum is at one of its extreme positions, the reference circle is on the x axis and the centripetal acceleration vector points back along the x axis toward the origin, so at this point the acceleration of the pendulum is equal to the centripetal acceleration of the reference point. • When the pendulum is at its equilibrium position the reference point lies directly above or below the pendulum, with its centripetal acceleration vector pointing in the vertical direction. The centripetal acceleration vector therefore has no component in the direction of the pendulum's motion, and the pendulum is for that instant not accelerating. • At any other position the centripetal force vector has a nonzero component in the x direction which has a magnitude less than that of the vector. This component is identical to the acceleration of the pendulum.** STUDENT QUESTION If we want to find the centripetal acceleration of a circle we can use the formula a_cent= v^2/rThe only thing I wonder about is if we would know what the radius of the circle is or is there a formula we could use to solve for this? INSTRUCTOR RESPONSE The radius of the reference circle is equal to the amplitude of motion. When a pendulum moving along the x axis it at its maximum position x = A, the reference point is passing through the positive x axis, and the r vector is directed along the x axis. Thus the magnitude of the r vector is equal to the amplitude A, and the radius of the circle is therefore equal to the amplitude. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!