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QuestionFind the minimum mass mm to place on a plank of length 6.31 m6.31 \mathrm{~m} and mass 16.86 kg16.86 \mathrm{~kg} for balance.

Studdy Solution

STEP 1

Assumptions1. The length of the plank is 6.31 m6.31 \mathrm{~m} . The mass of the plank is 16.86 kg16.86 \mathrm{~kg}
3. The length of the plank in contact with the boat is 1.95 m1.95 \mathrm{~m}
4. The mass of the person is 66.4 kg66.4 \mathrm{~kg}
5. The plank will not rotate if the torques on both sides of the pivot point (the edge of the boat) are equal. The torque due to the person and the plank on one side should be equal to the torque due to the mass on the other side.

STEP 2

First, we need to find the center of mass of the plank. This is located at the midpoint of the plank.
Center of mass of plank=Length of plank/2Center\, of\, mass\, of\, plank = Length\, of\, plank /2

STEP 3

Plug in the given value for the length of the plank to calculate the center of mass of the plank.
Center of mass of plank=6.31 m/2Center\, of\, mass\, of\, plank =6.31\, m /2

STEP 4

Calculate the center of mass of the plank.
Center of mass of plank=6.31 m/2=3.155 mCenter\, of\, mass\, of\, plank =6.31\, m /2 =3.155\, m

STEP 5

Now, we need to find the distance from the pivot point to the center of mass of the plank.
Distance to center of mass of plank=Center of mass of plank−Length of plank on boatDistance\, to\, center\, of\, mass\, of\, plank = Center\, of\, mass\, of\, plank - Length\, of\, plank\, on\, boat

STEP 6

Plug in the values for the center of mass of the plank and the length of the plank on the boat.
Distance to center of mass of plank=3.155 m−1.95 mDistance\, to\, center\, of\, mass\, of\, plank =3.155\, m -1.95\, m

STEP 7

Calculate the distance to the center of mass of the plank.
Distance to center of mass of plank=3.155 m−1.95 m=1.205 mDistance\, to\, center\, of\, mass\, of\, plank =3.155\, m -1.95\, m =1.205\, m

STEP 8

Now, we can calculate the torque due to the plank. Torque is the product of the force (which is mass times gravity) and the distance from the pivot point.
Torque due to plank=Mass of plank×Gravity×Distance to center of mass of plankTorque\, due\, to\, plank = Mass\, of\, plank \times Gravity \times Distance\, to\, center\, of\, mass\, of\, plank

STEP 9

Plug in the values for the mass of the plank, gravity, and the distance to the center of mass of the plank. We'll use 9.8 m/s29.8\, m/s^2 for gravity.
Torque due to plank=16.86 kg×9.8 m/s2×.205 mTorque\, due\, to\, plank =16.86\, kg \times9.8\, m/s^2 \times.205\, m

STEP 10

Calculate the torque due to the plank.
Torque due to plank=16.86 kg×9.8 m/s2×.205 m=198.8 N⋅mTorque\, due\, to\, plank =16.86\, kg \times9.8\, m/s^2 \times.205\, m =198.8\, N\cdot m

STEP 11

Next, we need to calculate the torque due to the person. The person is at the end of the plank, so the distance from the pivot point is the length of the plank minus the length of the plank on the boat.
Distance to person=Length of plank−Length of plank on boatDistance\, to\, person = Length\, of\, plank - Length\, of\, plank\, on\, boat

STEP 12

Plug in the values for the length of the plank and the length of the plank on the boat.
Distance to person=6.31 m−.95 mDistance\, to\, person =6.31\, m -.95\, m

STEP 13

Calculate the distance to the person.
Distance to person=6.31 m−.95 m=.36 mDistance\, to\, person =6.31\, m -.95\, m =.36\, m

STEP 14

Now, we can calculate the torque due to the person.
Torque due to person=Mass of person×Gravity×Distance to personTorque\, due\, to\, person = Mass\, of\, person \times Gravity \times Distance\, to\, person

STEP 15

Plug in the values for the mass of the person, gravity, and the distance to the person.
Torque due to person=66.4 kg×9.8 m/s2×4.36 mTorque\, due\, to\, person =66.4\, kg \times9.8\, m/s^2 \times4.36\, m

STEP 16

Calculate the torque due to the person.
Torque due to person=66.4 kg×9.8 m/s2×4.36 m=2835.2 N⋅mTorque\, due\, to\, person =66.4\, kg \times9.8\, m/s^2 \times4.36\, m =2835.2\, N\cdot m

STEP 17

The total torque on the side with the person and the plank is the sum of the torques due to the person and the plank.
Total torque=Torque due to person+Torque due to plankTotal\, torque = Torque\, due\, to\, person + Torque\, due\, to\, plank

STEP 18

Plug in the values for the torque due to the person and the torque due to the plank.
Total torque=2835.2 N⋅m+198.8 N⋅mTotal\, torque =2835.2\, N\cdot m +198.8\, N\cdot m

STEP 19

Calculate the total torque.
Total torque=2835. N⋅m+198.8 N⋅m=3034 N⋅mTotal\, torque =2835.\, N\cdot m +198.8\, N\cdot m =3034\, N\cdot m

STEP 20

Now, we can find the minimum mass needed on the end of the board on the ship. This mass must create a torque equal to the total torque on the other side. The distance from the pivot point to this mass is the length of the plank on the boat.
Mass needed=Total torque/(Gravity×Length of plank on boat)Mass\, needed = Total\, torque / (Gravity \times Length\, of\, plank\, on\, boat)

STEP 21

Plug in the values for the total torque, gravity, and the length of the plank on the boat.
Mass needed=3034 N⋅m/(9.8 m/s×1.95 m)Mass\, needed =3034\, N\cdot m / (9.8\, m/s^ \times1.95\, m)

STEP 22

Calculate the minimum mass needed on the end of the board on the ship.
Mass needed=3034 N⋅m/(9.8 m/s×1.95 m)=159.5 kgMass\, needed =3034\, N\cdot m / (9.8\, m/s^ \times1.95\, m) =159.5\, kgThe minimum mass you must put on the end of the board that is still on the ship so that you can successfully walk to the end of the plank without have the board rotate you into the water is 159.5 kg159.5 \mathrm{~kg}.

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