Tutor profile: Abhishek J.
How to find the shortest distance of a known point in 3D space with co-ordinates (a,b,c) to a circle resting in XY plane with center on the origin? Radius of the circle is know, say 'r'. (shortest distance between a point and a circle will be w.r.t. a point on the circumference of the circle). Hint: Shortest distance is always perpendicular distance and try to solve the question for only the First quadrant.
Well, there is no straight forward answer to this question, as most geometrical questions will be theoretical proofs. You start with realizing that there are 4 important points in this question: Origin 'A' with co-ordinates (0,0,0) Point 'C' in 3D space with co-ordinates (a,b,c) Point 'D' along the Z axis with co-ordinate (0,0,c) [it's at the same height as that of point 'C'] And finally point 'B' which is on the circle and closest to point 'C'. The co-ordinates of this point is (r*cos(alpha), r*sin(alpha), 0) as it is on a circle and on XY plane. Here 'alpha' is the unknown angle and we need to find it in order to find the distance. Now, the important factor to consider is that all the 4 points must lay in one plane for point 'B' to be closest to point 'C', here is the perpendicular hint coming into the picture. You can apply vector calculations to find that : tan(alpha) = b/a. This is how you can find co-ordinates of point 'B' and apply distance formula to get shortest distance!! P.S.: Trigonometric 'tan' takes care of other quadrants as well, so you can just solve for First quadrant.
Subject: Mechanical Engineering
Two identical cylinders are rolling on 2 identical ramps from state of rest. One ramp is smooth and second one has a ditch along the way (a smooth depression followed by a smooth elevation). Which of the 2 cylinders will reach the bottom of the ramp first? Hint: It's not always about the velocity but acceleration plays a key role!!
From a clear Kinematic velocity point of view: All potential energy at the top of the ramp is converted into Kinetic velocity as the cylinders move downward. So, starting with same height, i.e. same Potential Energy, the cylinders will have same Kinetic velocity, i.e. same velocity (cause the mass is same), at the bottom of the ramp. So, the cylinder in the ditch will first accelerate and then decelerate in the ditch, keeping the velocity before the ditch as same with the other cylinder and also the velocity after the ditch as same with the other cylinder. And since the velocity of either cylinders is constantly rising (cause the ramp is declined surface), the cylinder in the ditch moved faster in the ditch as compared to the smooth surface ramp cylinder. So, the cylinder in the ditch reached the bottom first!! There are youtube videos you can watch for this experiment, or you can make one at home. Just remember to keep this identical, and it should work fine.
How to create any number between 1 to 1,000 as addition of only 10 numbers. Hint: You can use those same 10 to go up to 1023
Welcome to binary!! understanding the power of binary numbers is the key to understanding how numbers, algebra, math and computers are all connected. Let's solve the above problem for instance: The 10 numbers you need are 2^0, 2^1, 2^2, 2^3, 2^4.....2^9. It becomes very self-explanatory when you actually see the number as 1, 2, 4, 8, 16...512. 1 & 2 can be added to form 3, 1 & 4 to get 5, 2 & 4 for 6, 1 2 & 4 add up to 7 and the list goes on till you add all the previous numbers to 512 to get 1023. This is just the tip of the iceberg to why binary numbers have found a place in computers and mathematics.