Physics 3210

Week 3 clicker questions

A particle of mass m slides on the outside of a cylinder of radius a. What condition must be satisfied by the force of constraint at the point (angle 0) where the particle leaves the cylinder?

a

A. Qr= mg

B. Qr= mg sin 0

C. Qr= 0

D. Qr= –mg

E. Qr= –mg sin 0

What can you conclude from the following relation?

A.

B.

C.

D.

ii i

x 0x

LL

i i

0x

L

i

0 for all ix

L

ix 0 for all i

r 0L

What can you conclude from the following relation?

A. Nothing.

B. Angular momentum is conserved (for a closed system).

C. Momentum is conserved (for a closed system).

D. Energy is conserved (for a closed system).

E. Lagrangian mechanics is fun.

i

constant for all ix

L

Which of these choices is a valid way to rewrite the following relation?

A.

B.

C.

D.

i ii i i

x x 0x x

L LL

i i i ii

p x p xL

i i i ii

p x p xL

i i i ii

p x p xL

i i i ii

p x p xL

How many of the following statements are true?

1. Momentum is conserved because space is homogeneous.2. Angular momentum is conserved because space is homogeneous.3. Angular momentum is conserved because space is isotropic.4. Momentum is conserved because space is isotropic.5. Energy is conserved because space is homogeneous and isotropic.

A. Exactly one

B. Exactly two

C. Exactly three

D. Exactly four

E. All five

Which of these choices is the correct expression for the time derivative of the Lagrangian of a closed system?

A. B.

C. D.

jj j

dq

dt q

L L

j jj j j

dq q

dt q q

L L L

jj j

dq

dt q

L L

j j

j j j

dq q

dt q q

L L L

Physics 3210

Week 3 - Wednesday clicker questions

How can we rewrite the following expression?

A. B.

C. D.

2

1

t

j j

t

dp q dt ?

dt

2

1

t

j j

t

q p dt 2

1

t

j j

t

p q dt

2

1

t

j j

t

q p dt 2

1

t

j j

t

q p dt

If we know that

what can we say about the terms in parentheses?

A.

B.

C.

D.

2

1

t

j j j jj j jt

H Hp q q p dt 0

q p

j jj j

H HBoth p and q vanish independently.

q p

j jj j

H HThe sum p + q vanishes.

q p

j jj j

H HEither p or q vanishes, but not both.

q p

jj j

j j j

pH HThe ratio p q = .

q p q/

Physics 3210

Week 3 - Friday clicker questions

There has been a request to have the homework due Thursday (instead of Wednesday) to avoid conflicts with other classes. Which would you prefer?

A. Change to Thursday.

B. Stick with Wednesday.

C. I don’t care.

There has been a request to change the course grading so that the lowest of the 3 midterm exam grades is dropped. Which would you prefer?

A. Count all 3 exams.

B. Drop the lowest of the 3 exams.

C. I don’t care.

Hamilton’s canonical equations are:

If the Hamiltonian is independent of pj, which of the following is true?

A.

B.

C.

D.

jj

jj

Hq

p

Hp

q

jq is constant

jq is zero

jp is constant

jp is zero

A particle slides on a helical wire defined by z=kr=const. If we use as the generalized coordinate, what is the generalized momentum p?

A.

B.

C.

D.

2 2 21p m r k

2

p m r k

2 2p m r k

p m

A particle slides on a helical wire defined by z=kr=const. If we use as the generalized coordinate, the generalized momentumWhat is the Hamiltonian of the system?

A.

B.

C.

D.

2pH mgk

2m

2 2p m r k .

2pH mgk

2m

2pH mgk

2m r k

2

2 2

pH mgk

2m r k

An Atwood machine has total free-hanging length of string l connected to two masses m1 (on the left) and m2 (on the right). Use x as a generalized coordinate. What is the generalized momentum p?

A.

B.

C.

D.

21 2

1p m m x

2 m1

m2

x

l-x

1 2p m m x

1 2p m m x

1p m x

An Atwood machine has a Hamiltonian:

What are Hamilton’s equations of motion?

A.

B.

C.

1 2

1 2

px

m m

p m m g

m1

m2

x

l-x

2

1 2 21 2

pH m m gx m gl

2 m m

1 2

1 2

px

m m

p m m g

D.

E.

1 2

1 2

px

m m

p m m g

1 2

1 2

px

m m

p m m g

1 2

1 2

px

m m

p m m g

An Atwood machine has the following Hamiltonian equations of motion:

How does the system move?

A. x moves with constant gravitational acceleration with effective mass

B. x moves with constant gravitational acceleration with effective mass

C. x moves with constant gravitational acceleration with effective mass

D. x moves with constant velocity

E. x doesn’t change

m1

m2

x

l-x

1 2

1 2

px

m m

p m m g

1 2m m

1 2m m

1 2 1 2m m m m/