A 0.109-kg remote control 16.2 cm long rests on a table, as shown in the figure
ID: 2153465 • Letter: A
Question
A 0.109-kg remote control 16.2 cm long rests on a table, as shown in the figure below, with a length L overhanging its edge. To operate the power button on this remote requires a force of 0.305 N. How far can the remote control extend beyond the edge of the table and still not tip over when you press the power button? Assume the mass of the remote is distributed uniformly, and that the power button is on the end of the remote overhanging the table.
A 0.109-kg remote control 16.2 cm long rests on a table, as shown in the figure below, with a length L overhanging its edge. To operate the power button on this remote requires a force of 0.305 N. How far can the remote control extend beyond the edge of the table and still not tip over when you press the power button? Assume the mass of the remote is distributed uniformly, and that the power button is on the end of the remote overhanging the table.Explanation / Answer
asic idea: Torque from weight on table = Torque from weight off table + torque from force pushing button. don't forget the "arm" for the weight is 1/2 the distance so for the part off the table it is (L/2). the arm for the force is L ============================ Given (and taken... from wiki): ? the length overhanging: L cm ? the length NOT overhanging: (21.2 - L) cm ? the Force: 0.285 N ? the remote's TOTAL mass: 0.094 kg ? standard gravity: g ˜ 9.807 ? (The remote's TOTAL weight: (mass•g): ˜ 0.094 • 9.807 N ============================ since the remote is measured in "cm," I'm just going to use "cm" in the calculations so no conversions will be needed. ============================ Torque balancing the remote ON the table: Force • distance or Remote-Force(table-part) • arm(distance from edge to the point where the Remote-Force is acting) ? but only part of the remote is over the table! OK, fine. What fraction? (the length NOT overhanging) / (total length) or (21.2 - L) / (21.2) So, (21.2 - L) / (21.2) of the Remote-Weight is going to be the Remote-force helping to balance it. Remote-force: (0.094 • 9.807) • (21.2 - L) / (21.2) N and since the weight is uniformly distributed... (yeah, right... everyone knows the battery end is heavier) ...the weight acts at the midpoint. So, in this case, the Remote-force over the table acts at a distance that is ½ of the length NOT overhanging. the arm: ½ • (21.2 - L) cm So, the torque balancing the remote over the table is: Force • arm or ( (0.094 • 9.807) • (21.2 - L) / (21.2) ) • ( ½ • (21.2 - L) ) Ncm That's one side. ============================ The other side hanging off the table: (similar to above) the Remote-force on this side is the part of the remote's weight hanging OFF the table. OK, what fraction is that? (L / 21.2) So, the Remote-force hanging OFF is: (0.094 • 9.807) • ( L / 21.2 ) N the arm is obviously ½ of the distance hanging OFF the table: (since the weight is going to act in the middle) (½ • L) cm So, the torque from the Remote-force is: ( (0.094 • 9.807) • ( L / 21.2 ) ) • (½ • L) Ncm ? But, there is the torque from Button-Pushing that is: Force • distance or (0.285 • L) Ncm ============================ To balance the remote, the torque on one side must equal the torques on the other: ( (0.094 • 9.807) • (21.2 - L) / (21.2) ) • ( ½ • (21.2 - L) ) = (0.285 • L) + ( (0.094 • 9.807) • ( L / 21.2 ) ) • (½ • L) ============================ Let's multiply by 2 to get rid of the ½ and multiply by 21.2 to get rid of 21.2 on the bottom.: Now, just solve for L: ( (0.094 • 9.807) • (21.2 - L) ) • (21.2 - L) = (2 • 21.2 • 0.285 • L) + ( (0.094 • 9.807) • L ) • L or (0.094 • 9.807) • ( (21.2)² - (2 • 21.2 • L) + L² ) = (2 • 21.2 • 0.285 • L) + (0.094 • 9.807) • L² or (subtracting the L² terms on both sides) (0.094 • 9.807) • ( (21.2)² - (2 • 21.2 • L) ) = (2 • 21.2 • 0.285 • L) or (put the L terms on one side) (0.094 • 9.807) • (21.2)² = (0.094 • 9.807) • (2 • 21.2 • L) + (2 • 21.2 • 0.285 • L) or (0.094 • 9.807) • (21.2)² = (2 • 21.2 • L) • ( (0.094 • 9.807) + 0.285) ) or ( (0.094 • 9.807) • (21.2) ) / (2 • ( (0.094 • 9.807) + 0.285) ) ) = L or ????????? ??L ˜ 8.097 cm? ????????? (which kind of makes sense... most of the remote has to be over the table since you are pushing down on the part that is hanging off the table)
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