course Phy 231
......!!!!!!!!...................................
13:40:06 Set 4 probs 1-7 If we know the net force acting on an object and the time during which the force acts, we can find the change in what important quantity?
......!!!!!!!!...................................
RESPONSE --> Change in momentum (mv)
.................................................
......!!!!!!!!...................................
13:40:14 ** You can find the change in the momentum. Fnet * `ds is change in KE; Fnet * `dt is change in momentum. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
13:40:39 What is the definition of the momentum of an object?
......!!!!!!!!...................................
RESPONSE --> momentum = mass * velocity p = mv
.................................................
......!!!!!!!!...................................
13:40:56 ** momentum = mass * velocity. Change in momentum is mass * change in velocity (assuming constant mass). UNIVERSITY PHYSICS NOTE: If mass is not constant then change in momentum is change in m v, which by the product rule changes at rate dp = m dv + v dm. If mass is constant `dm = 0 and dp = m dv so `dp = m * `dv. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
13:41:18 How do you find the change in the momentum of an object during a given time interval if you know the average force acting on the object during that time interval?
......!!!!!!!!...................................
RESPONSE --> Fave * 'dt = 'dp
.................................................
......!!!!!!!!...................................
13:41:27 ** Since impulse = ave force * `dt = change in momentum, we multiply ave force * `dt to get change in momentum. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
13:42:40 How is the impulse-momentum theorem obtained from the equations of uniformly accelerated motion and Newton's Second Law?
......!!!!!!!!...................................
RESPONSE --> substitute a = Fnet/m (Newton's 2nd law) into vf = v0 + a'dt to get Fnet * 'dt = mvf - mv0
.................................................
......!!!!!!!!...................................
13:42:54 ** First from F=ma we understand that a=F/m. Now if we take the equation of uniformly accelerated motion vf= v0 + a'dt and subtract v0 we get vf-v0 = a'dt. Since vf-v0 = 'dv, this becomes 'dv = a'dt. Now substituting a=F/m , we get 'dv = (F/m)'dt Multiplying both sides by m, m'dv = F'dt **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
13:45:43 If you know the (constant) mass and the initial and final velocities of an object, as well as the time required to change from the initial to final velocity, there are two strategies we can use to find the average force exerted on the object. What are these strategies?
......!!!!!!!!...................................
RESPONSE --> 1. Calculate avg. acceleration aAve = (vf-v0)/'dt, then Fave = ma 2. Use impulse-momentum: 'dp = mvf - mv0, then set equal to impulse Fave*'dt and solve for Fave = 'dp/'dt
.................................................
......!!!!!!!!...................................
13:45:48 ** The impulse-momentum theorem for constant masses is m `dv = Fave `dt. Thus Fave = m `dv / `dt. We could alternatively find the average acceleration aAve = (vf - v0) / `dt, which we then multiply by the constant mass to get Fave. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
13:46:59 Class notes #14. How do we combine Newton's Second Law with an equation of motion to obtain the definition of energy?
......!!!!!!!!...................................
RESPONSE --> Substitute a = F/m into vf^2 = v0^2 + 2a'ds and solve to get F'ds = .5 mvf^2 - .5 mv0^2
.................................................
......!!!!!!!!...................................
13:47:08 ** a = F / m. vf^2 = v0^2 + 2 a `ds. So vf^2 = v0^2 + 2 (Fnet / m) `ds. Multiply by m/2 to get 1/2 mvf^2 = 1/2 m v0^2 + Fnet `ds so Fnet `ds = 1/2 m vf^2 - 1/2 m v0^2--i.e., work = change in KE. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
13:49:33 What is kinetic energy and how does it arise naturally in the process described in the previous question?
......!!!!!!!!...................................
RESPONSE --> kinetic energy is energy of motion, equal to .5 mv^2, and the previous question defines change in kinetic energy to be equal to the work done, or the force applied over a given distance
.................................................
......!!!!!!!!...................................
13:49:38 ** KE is the quantity 1/2 m v^2, whose change was seen in the previous question to be equal to the work done by the net force. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
13:50:42 What forces act on an object as it is sliding up an incline?
......!!!!!!!!...................................
RESPONSE --> Weight of the object Normal force perpendicular to the incline Frictional force in the direction of the incline
.................................................
......!!!!!!!!...................................
13:51:24 ** Gravitational force can be broken into two components, one parallel and one perpendicular to the ramp. The normal force exerted by the ramp is an elastic force, and unless the ramp breaks the normal force is equal and opposite to the perpendicular component of the gravitational force. Frictional force arises from the normal force between the two surfaces, and act in the direction opposed to motion. The gravitational force is conservative; all other forces in the direction of motion are nonconservative. COMMON ERROR: The Normal Force is in the upward direction and balances the gravitational force. COMMENT: The normal force is directed only perpendicular to the incline and is in the upward direction only if the incline is horizontal. The normal force cannot balance the gravitational force if the incline isn't horizontal. Friction provides a component parallel to the incline and opposite to the direction of motion. **
......!!!!!!!!...................................
RESPONSE --> Friction is opposite to the direction of motion. Normal force is opposite to the component of the weight perpendicular to the incline.
.................................................
......!!!!!!!!...................................
13:52:36 For an object sliding a known distance along an incline how do we calculate the work done on the object by gravity? How do we calculate the work done by the object against gravity?
......!!!!!!!!...................................
RESPONSE --> Take the component of the force of gravity parallel to the incline and multiply it by the distance to determine work done by gravity on the object. The work done by the object against gravity is equal and opposite.
.................................................
......!!!!!!!!...................................
13:55:06 ** The gravitational force is m * g directly downward, where g is the acceleration of gravity. m * g is the weight of the object. If we know change in vertical position then we can simply multiply weight m * g with the vertical displacement `dy, being careful to keep track of which is positive and/or negative. Alternatively it is instructive to consider the forces in the actual direction of motion along the incline. For small inclines the component of the gravitational force which is parallel to the incline is approximately equal to the product of the weight and the slope of the incline, as seen in experiments. The precise value of the component parallel to the incline, valid for small as well as large displacements, is m g * sin(theta), where theta is the angle of the incline with horizontal. This force acts down the incline. If the displacement along the incline is `ds, measured with respect to the downward direction, then the work done by gravity is the product of force and displacement, m g sin(theta) * `ds. If `ds is down the incline the gravitational component along the incline is in the same direction as the displacement and the work done by gravity on the system is positive and, in the absence of other forces in this direction, the KE of the object will increase. This behavior is consistent with our experience of objects moving freely down inclines. If the displacement is upward along the incline then `ds is in the opposite direction to the gravitational force and the work done by gravity is negative. In the absence of other forces in the direction of the incline this will result in a loss of KE, consistent with our experience of objects coasting up inclines. The work done against gravity is the negative of the work done by gravity, positive for an object moving up an incline (we have to use energy to get up the incline) and negative for an object moving down the incline (the object tends to pick up energy rather than expending it) **
......!!!!!!!!...................................
RESPONSE --> This is much more complete. It is also good to remember that gravitational potential energy = m*g*'dy
.................................................
......!!!!!!!!...................................
13:56:40 For an object sliding a known distance along an incline how do we calculate the work done by the object against friction? How does the work done by the net force differ from that done by gravity?
......!!!!!!!!...................................
RESPONSE --> The work done against friction is the coefficiant of friction times the normal force multiplied by the distance. This differs from the force done by gravity in that it is opposite, and also not equal unless the incline is horizontal, because the normal force balances the component of the gravitational force that is perpendicular to the incline.
.................................................
......!!!!!!!!...................................
13:57:33 ** The work done against friction is the product of the distance moved and the frictional force. Since the force exerted by friction is always opposed to the direction of motion, the force exerted by the system against friction is always in the direction of motion so the work done against friction is positive. The net force on the system is sum of the gravitational component parallel to the incline and the frictional force. The work done by the net force is therefore equal to the work done by gravity plus the work done by the frictional force (in the case of an object moving up an incline, both gravity and friction do negative work so that the object must do positive work to overcome both forces; in the case of an object moving down an incline gravity does positive work on the system while friction, as always, does negative work on the system; in the latter case depending on whether the work done by gravity on the system is greater or less than the frictional work done against the system the net work done on the system may be positive or negative) **
......!!!!!!!!...................................
RESPONSE --> Yes, and work done against friction is positive, right. I didn't think of the fact that the net force on the system is that sum.
.................................................
......!!!!!!!!...................................
14:01:27 Explain why the restoring force on a simple pendulum is in nearly the same proportion to the weight of the pendulum as its displacement from equilibrium to its length, and explain which assumption is made that makes this relationship valid only for displacements which are small compared to pendulum length.
......!!!!!!!!...................................
RESPONSE --> I know the equation F = -k * x, but I'm not sure I understand why the proportion of the restoring force to the weight is nearly the same as that of its displacement from equilibrium to its length.
.................................................
......!!!!!!!!...................................
14:02:20 ** In terms of similar triangles: The reason the approximation only works for small displacements is because the sides used on one triangle are not the same as the sides used on the other. From the triangle we see that the restoring force and the weight are at right angles, while the length and horizontal displacement of the pendulum from equilibrium are the hypotenuse and the horizontal leg of a triangle and hence are not at right angles. For small angles the two long sides of the triangle are approximately equal so the discrepancy doesn't make much difference. For larger angles where the two long sides are significantly different in length, the approximation no longer works so well. In terms of components of the vectors: The tension force is in the direction of the string. The component of the tension force in the horizontal direction is therefore seen geometrically to be in the same proportion to the total tension as the length of the pendulum to the horizontal displacement (just draw the picture). The vertical component of the tension force must be equal to the weight of the pendulum, since the pendulum is in equilibrium. If the displacement is small compared to the length the vertical component of the tension force will be very nearly equal to the tension force. So the previous statement that 'The component of the tension force in the horizontal direction is therefore seen geometrically to be in the same proportion to the total tension as the length of the pendulum to the horizontal displacement' can be replaced by the statement that 'The component of the tension force in the horizontal direction is therefore seen geometrically to be in the same proportion to the weight of the pendulum as the length of the pendulum to the horizontal displacement. **
......!!!!!!!!...................................
RESPONSE --> I need to research this and learn it.
.................................................
17:07:42 **** Univ: 6.58 (6.50 10th edition). chin-up .40 m, 70 J/kg of muscle mass, % of body mass in pullup muscles of can do just 1. Same info for son whose arms half as long.
......!!!!!!!!...................................
RESPONSE --> a) W = (9.8 m/s^2)(1 kg)(.40 m) = 3.92 J b) 3.92 J / 70 J = 5.6% c) W = (9.8 m/s^2)(1 kg)(.20 m) = 1.96 J 1.96 J / 70 J = 2.8% Children apply the same amount of force over a smaller distance, therefore perform less work, thus the chinups are easier.
.................................................
......!!!!!!!!...................................
17:07:52 ** For each kg of mass the weight is 1 kg * 9.8 m/s^2 = 9.8 N. Work done to lift each kg of mass .4 m would then be 9.8 N * .4 m = 3.92 J. The chin-up muscles generate 3.92 J per kg, which is 3.92 / 70 of the work one kg of muscle mass would produce. So the proportion of body mass in the pullup muscles is 3.92 / 70 = .056, or 5.6%. For the son each kg is lifted only half as far so the son only has to do half the work per kg, or 1.96 J per kg. For the son the proportion of muscle mass is therefore only 1.96 / 70 = 2.8%. The sons advantage is the fact that he is lifting his weight half as high, requiring only half the work per kg. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
17:18:25 Univ. 6.72 (6.62 10th edition). net force 5 N/m^2 * x^2 at 31 deg to x axis; obj moves along x axis, mass .250 kg, vel at x=1.00 m is 4.00 m/s so what is velocity at x = 1.50 m?
......!!!!!!!!...................................
RESPONSE --> First, find initial kinetic energy: kE0 = .5 (.250 kg)(4.00 m/s)^2 = 2.00 J W = integral(x1 to x2) Fx dx Fx = 5.00 N/m^2 cos 31deg = 4.29 N/m^2 W = 4.29 N/m^2 * integral(1.00m to 1.50m) x^2 dx = 4.29 N/m^2(1/3*1.50m^3 - 1/3*1.00m^3) = 3.40 J kEf = kE0 + W = 2.00 J + 3.40 J = 5.40 J vf^2 = 5.40 J / (.5 * .250 kg) vf = 6.57 m/s
.................................................
......!!!!!!!!...................................
17:21:30 ** Force is variable so you have to integrate force with respect to position. Position is measured along the x axis, so you integrate F(x) = - k / x^2 with respect to x from x1 to x2. An antiderivative of - k / x^2 is k / x so the integral is k / x2 - k / x1. If x2 > x1, then k / x2 < k / x1 and the work is negative. Also, if x2 > x1, then motion is in the positive x direction while F = - k / x^2 is in the negative direction. Force and displacement in opposite directions imply negative work by the force. For slow motion acceleration is negligible so the net force is practically zero. Thus the force exerted by your hand is equal and opposite to the force F = - k / x^2. The work you do is opposite to the work done by the force so will be - (k / x2 - k / x1) = k/x1 - k/x2, which is positive if x2 > x1. This is consistent with the fact that the force you exert is in the opposite direction to the force, therefore in the positive direction, as is the displacement. Note that the work done by the force is equal and opposite to the work done against the force. **
......!!!!!!!!...................................
RESPONSE --> This is the answer to 6.64, not 6.72 - however I did have questions on that problem and this has helped. Was I correct on 6.72?
.................................................