Galileo's Inclined PlaneGalileo attempted to determine free fall acceleration on Earth by "slowing it down." He constructed inclined planes,down which he observed discs to roll.
He measured time versus distance of travel to calculate Earth's gravity, go. On one test (geometry as shown) Galileo used powder on the ramp and a small cube of steel. The time he measured was 2.4 seconds. Assume the cube to slide down the plane without friction. What value of go does this experiment predict?
In his life, he had attempted to determine μME by "slowing" the effect of gravity on free-fall of a body. He studied the motion of bodies moving down inclined planes. Specifically, he measured time versus distance-of-travel and used that invormation to calculate Earth's gravity, go,E. For one test (geometry as shown) he used a small, highly-polished cube of steel. He observed the cube to slide down the incline a distance of 216 centimeters in 2 seconds. What value of go,E did Galileo's determine?
♦ This discussion is a bit advanced (in the direction of learning then understanding). We will use "proper math" (vectors) with a strategem to deal with "an inclined plane."
This will be a special coordinate system (0'X'Z') with the X'-axis parallel with the plane (and increasing "up the plane"). The unit vectors are then I' and K'. The free-body diagram shows this information. The momentum equation and its forces are written below:
We proceed carefully by writing the normal and gravity force vectors in component form (above), then placing them in the momentum equation (second law) below. Motion involves the "up or down hill" the I' component. Hence we scalar multiply by the vector direction - I '. Recall that I '• I ' equals 1 and I ' • K ' equals 0.
So here we are again. The above first order differential equation tells us the change of momentum of the block is a constant. Initially the velocity of the block is zero and we place the beginning of the coordinate space (0X'Z') to be the beginning position. The relevant space for the slide is 0X' and the - on the term right of equalty means the block moves leftward, down the plane. Though Galileo had no calculus, he knew the velocity and position of the block to be.
Applying the numbers of his test into the equation (above right) we obtain:
