The diagrams above show an astronaut in a sealed room (with air). The astronaut is weighing a pink vase under three conditions. In diagram A, the room is on the surface of the Earth. In diagram B, the room is floating in space (far from planets, stars, and other gravitational bodies) so that there is effectively zero gravity. In diagram C, the room is also far from gravitational bodies but the room is accelerating upward at 9.8 m/s2.
In diagram B, the scale would read that the weight of the vase is zero. In both diagrams A and C, the scale would read the weight of the vase on the surface of the Earth. In situations A and C, the astronaut would feel exactly the same (with a force equal to the weight of the astronaut between her feet and the floor). The central idea of Einstein's General Theory of Relativity is that there is no experiment the astronaut could do to determine whether she was in situation A or situation C. The two situations are identical. A tremendous amount of evidence corroborating Einstein's idea has been compiled in the last century.
In diagram A, the scale measures the gravitational mass of the vase. On the surface of the Earth, the weight of an object is its gravitational mass multiplied by 9.8 m/s2. In diagram C, the scale is measuring the inertial mass of the vase. The scale would read zero if the vase had no inertia, but since it does have inertia the scale reads the inertial mass of the vase multiplied by its acceleration which is 9.8 m/s2 in diagram C. Another way of stating the central idea of General Relativity is that gravitational and inertial mass are exactly the same. We know of no fundamental reason why this has to be true, but it seems to be true.
The questions that follow explore the effect of inertial mass on the fuzzy dice that your friend has hanging from the rearview mirror of his car. [Note: Air Force pilots in WWII began hanging real dice from the windows of their fighter planes for good luck. That idea evolved into fuzzy dice in automobiles after the war.]
NOTE: Cant really show pictures but they are described
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To answer the questions below, it may be useful to think of your friend's car driving on a level road on the surface of the Earth, or maybe in space accelerating upwards at 9.8 m/s2 (or some other rate of acceleration, depending on the question).
1) Your friend starts out by hanging is fuzzy dice from a spring. On the surface of the Earth, he finds the length of the spring to be 8.4 cm. With his car drifting in space (as in diagram B, above) he finds the length of the spring to be 3.5 cm. What would be the length of the spring in a situation similar to diagram C above, if the car were accelerating upward at a rate of 9.8 m/s2?
2) What would be the length of the spring in a situation similar to diagram C above if the car were accelerating upward at a rate of 12.4 m/s2?
3)What would be the length of the spring in a situation similar to diagram C above if the car were accelerating upward at a rate of 6.6 m/s2?
Note for clarification: Answer all 3 questions since they correspond with each other. and make sure to "bold" the answers somehow. Also just for heads up #1 is NOT 13.3 cm and #2 is NOT 14.6 cm and #3 is not 11.7 cm they were incorrect so don't right those as the answers please. Furthermore, if your answer is wrong then be able to respond back quickly.
To answer these questions, we need to understand the relationship between the length of the spring and the acceleration of the car. The change in length of the spring indicates a change in the force acting on it due to the acceleration.
We can use Newton's second law of motion, which states that force is equal to mass multiplied by acceleration (F = ma). In this case, the force acting on the spring is related to its length.
Let's assume the initial length of the spring is 8.4 cm when the car is on the surface of the Earth (as mentioned in question 1). The force acting on the spring will be equal to the weight of the spring.
1) In diagram B, where the car is floating in space with effectively zero gravity, the force acting on the spring is zero. Therefore, the length of the spring will be its initial length, which is 8.4 cm.
2) In diagram C, where the car is accelerating upward at a rate of 12.4 m/s^2, we can calculate the force acting on the spring using the formula F = ma.
Since the force acting on the spring is equal to its weight, we can equate the force to the gravitational mass (m_g) of the spring multiplied by the acceleration due to gravity (g = 9.8 m/s^2). Therefore, we have:
F = m_g * g
We can rearrange this equation to solve for the gravitational mass:
m_g = F / g
Now, we can use the obtained gravitational mass to calculate the force acting on the spring in diagram C:
F = m_g * (acceleration of the car in diagram C)
And finally, by comparing the force in diagram C to the force in diagram A, we can find the ratio of the lengths of the spring between the two scenarios. This ratio will allow us to find the length of the spring in diagram C.
3) We can follow the same steps as in question 2, but with a different rate of acceleration (6.6 m/s^2) for the car in diagram C.
By applying these principles, we can now calculate the length of the spring in the situations similar to diagram C for each question.