1. A 30kg child is swinging on a swing whose seat is a distance of 5.0m from the pivot point. Estimate the optimal time between “pumps” that the child should execute to increase her swinging amplitude. How does this time change for a 15.0 kg child?

• (11 got this right, 3 made a slight error, 13 didn’t answer and 12 didn’t know what to do).

• My best guess is to use the equation T= 2(pi)*square root of (I/mgL). However, we dont have the rotational inertia of the pendulum (I). However, it is apparent that if you change the mass, it will affect the time between ˜pumps˜.

• use equation T=2piesqrootI/mgh so I=mLsqrd and the optimal time T=4.48sec so the finally answer for a 15kg child changes to T=4.49s i was not for sure how to plug in this data especially for I (for someone who did not know what to, this is VERY GOOD!)

• T = 2*pi*(L/g)^(1/2) The time does not change. The estimate time is 4.5 s.

Physical Pendulum

I = mghsin() but if <<1 then: = (mgh/I)

Be careful to distinguish the physical angle from the phase angle here!

Chapter 15 Problems

Chapter 15 problems

Rotational Inertia for Selected objects and rotation axes: HR&W Table 10-2

Review for Exam III• Rotational dynamics and kinematics:

t+ot+½ t2 2=o2 + 2(o)

I = miri2 = r x F = I

Be careful in relating linear to rotational motion. Ang. momentum: l = rxp L=IdL/dt= Statics: analysing/identifying torques and forces

Be careful about signs, chose your pivot points wisely Gravity F=G(m1m2/r12

2) Ug =-G(m1m2/r12) Kepler’s laws: Constant areas, T2= 42/GM a3

Fluid Statics: P=Po+gh Archemdes’ and Pascal’s principles

Fluid Dynamics: A1v1 = A2v2 P +gy + ½ v2 = const.

Review for Exam III (cont.)• Oscillations:

a= d2x/dt2 = -(k/m)[x(t) –xo]=> x(t)-xo = A cos(t

k/m

g/l

mgl/I T = 2

Exam is 4 MC questions followed by 3 problems (3 parts each). Total 78 Points.

Chapter 12 problems• =30 deg

• Mp = 10.2 kg

• Mb = 51.0 kg

Chapter 13 problems

Chapter 14 problems

•Pressures must be equal, but areas are not. The force from the spring is 1.50kN (=20kN/m*0.05 m), so the weight of the sand must be 83.3 N (=1.5kN/18.0) so the mass of sand is 8.5kg.

Chapter 15 problems

P221 Exam 3 Histogram

02468

1012

30 35 40 45 50 55 60 65 70 75 80

Bin

Freq

uenc

y

Frequency

Average: 51.2/78 = 65.6 %

• Y(x,t) = A sin( kx - ω t + φ)Assuming that x, y and t are all given in SI units, what are the appropriate units for A, k, ω, and φ? If these quantities are all positive, in what direction is the above wave traveling? [Units were no problem, but the direction proved more difficult for people to describe]

• The wave would be traveling at a diagonal up and to the right.

• If all the quantities are positive, that means the displacement is also positive, which means the wave is moving up the y-axis.

• K: rad/m w: rad/s φ: rad ? traveling in the positive direction [What exactly do you mean, and how do you really tell?]

Traveling waves

Chapter 16 Problems

Provide as concise a definition for the phase angle of a wave as you can manage.

• The phase constant is a value that shifts a curve to give a different displacement and slope at time t=0. (8)

• The phase angle of a point on a wave is the distance between the point and some other reference point on the wave. This distance is given in angular measurements (8)

• the book definition is on page 417 but i'm not really sure what it means. well, the phase of a wave is given in a equation (kx-wt) of the sine. the phase and sine function gives the oscillation of a string element.????

• An angle that changes linearly with time and position and that describes the point within a period at which an oscillating system finds itself at a particular place and time (the argument of the sinusoid).