9.1.4 Length Contraction

Question 1

Length Contraction

Answer 1

• Like time, length (or distance) is a relative quantity. Observers in different reference frames disagree on the length of an object.
• To measure the length of a moving object, you must note the position of its ends at the same time. The fact that simultaneity is relative leads to the phenomenon of length contraction.
• The length of a moving object is shortened, or contracted, when compared to the length of that object at rest (the proper length):

Question 2

A physicist G measures the height of the atmosphere, starting from the ground to the top of the atmosphere where muons are created. In G’s reference frame, the height of the atmosphere is 10 km. Imagine a physicist, M, moving along a muon with speed 0.9c. Which of the following is the height of the atmosphere measured by M ?

Answer 2

4.4 km

Question 3

Two rods A and B have the same proper length in a given reference frame. Suppose the two rods are moving at speeds (1/3) c and (2/3) c, respectively, relative to an observer O at rest. Which of the following accurately compares the length of rod A and rod B as measured by O ?

Answer 3

lA > lB

Question 4

If L0 is the proper length in the horizontal direction (left-right) of an object as measured by observer O, which of the following is the observer’s state of motion relative to the object?

Answer 4

The observer is at rest relative to the object.

Question 5

The proper length and width of a rod are l0 and w0, respectively. To an observer O, the rod is moving to the right with velocity v, as shown in the figure. Which of the following is the length l and width w of the rod measured by O?

Answer 5

l = l0
w = w0/g

Question 6

Which of the following is the correct procedure for measuring the length of an object in a given reference frame?

Answer 6

The positions of both the ends are measured simultaneously.

Question 7

A muon covers roughly 10 km and reaches the ground, as measured by a ground physicist G. The life of the muon is 2.210^-6 s for a moving physicist M, who observes the muon at rest. A muon traveling at 0.9 c will cover (0.9)(310^8 m/s)(2.2*10^-6s) = 534 m = 0.53 km. Which of the following correctly explains this contradictory result?

Answer 7

Physicist G observes the muon cover 10 km and reach the ground. For M, the height of the atmosphere has shrunk by 10/g km. This doesn’t contradict the experimental fact that the muon reaches the ground. Although G and M calculate a different height of the Earth’s atmosphere, both measurements are correct in their respective reference frames

Question 8

True or false?
An observer O measures the length d of a rod by measuring its left and right end positions simultaneously at time t. The rod is at rest relative to O. An observer O ′ moving to the right relative to the rod will also measure the same length d by measuring the positions of the two ends simultaneously at the same time t.

Answer 8

false

Question 9

Suppose delta l0 is the proper length of a rocket. Which of the following is the length of the rocket as measured by an observer who sees the rocket whiz by with velocity v?

Answer 9

delta l = delta l0 / g

Question 10

A rocket ship, with proper length l0, moves with a speed v relative to an observer. How fast must the rocket move for the observer to measure half of the rocket’s proper length?

Answer 10

0.866c