A 63 canoeist stands in the middle of her canoe. The canoe is 3.0 long, and the end that is closest to land is 2.6 from the shore. The canoeist now walks toward the shore until she comes to the end of the canoe. Suppose the canoeist is 3.3 from shore when she reaches the end of her canoe.
What is the canoe's mass?
better to sum torques about the center of mass.
from standing in the middle of the canoe we know that the CM is at x = 1.5+2.6 = 4.1
when the person is 3.3m from the shore that makes them (4.1m-3.3m)=0.8m from the CM and because we know the canoe is 3m long the CM of the canoe is (3.3+1.5)=4.8m from the shore and consequently (4.8m-4.1m)=0.7m from the CM of the system.
now sum the torques about the CM and set equal to zero with CCW rotation as being positive we get
0=(63kg)*(0.8m)-(Mcanoe)*0.7m
Mcanoe=(0.8/0.7)*63kg
To determine the mass of the canoe, we need to use the principle of center of mass. The center of mass of an object is the point at which all its mass can be considered to be concentrated.
In this scenario, the canoeist is standing in the middle of the canoe, which means the center of mass of the canoe and the person is also in the middle.
To calculate the mass of the canoe, we can use the concept of torque. Torque is the product of force and distance from a reference point. In this case, the reference point is the end of the canoe closest to the shore.
Let's assume that the mass of the canoe is "M" and the distance of the canoeist from the center of mass (midpoint of the canoe) is "d". The torque exerted by the canoeist can be calculated using the formula:
Torque = Force x Distance
The force exerted by the canoeist is equal to the weight of the canoeist, which is given by the formula:
Force = mass x gravitational acceleration
Since the gravitational acceleration is approximately 9.8 m/s^2, the force exerted by the canoeist is:
Force = 63 kg x 9.8 m/s^2
Now, we need to determine the distance "d" from the center of mass to the end of the canoe closest to the shore.
Since the total length of the canoe is 3.0 m, and the distance from the shore to the end of the canoe closest to the shore is 2.6 m, we can calculate "d" using the formula:
d = (Total length - Distance from the shore to the end closest to the shore) / 2
d = (3.0 m - 2.6 m) / 2
Finally, we can calculate the torque exerted by the canoeist using the formula:
Torque = Force x Distance
Substituting the values we have:
Torque = (63 kg x 9.8 m/s^2) x [(3.0 m - 2.6 m) / 2]
Now, solve for the torque and equate it to the torque required to maintain equilibrium:
Torque = M x gravitational acceleration x [(3.0 m - 2.6 m) / 2]
Simplifying the equation, we can solve for the mass of the canoe (M):
M = (Torque / (gravitational acceleration x [(3.0 m - 2.6 m) / 2]))
Substituting the known values, you can determine the mass of the canoe.
Well, the cg did not move.
The cg is (1.5+2.6) from shore at the start.
cg*totalmass= 3.3*63+masscanoe*(3.3+1.5)
(1.5+2.6)(63+masscanoe)= 3.3*63+masscanoe(4.8)
masscanoe(4.1-4.8)=3.3*63-4.1**63
solve for mass canoe