A girl standing on a stationary uniform plank 10m long. The plank can move with negligible friction along the surface of the water.
A) the girl walks along the plank with a velocity of 2ms-1 relative to the plank until she reaches the end of the plank. The centre of mass of the system is now 2m from the end of the plank. Calculate the mass of the girl is the mass of the plank is 12.6kg.
The center of mass of the system continues at rest because there are no external forces on the system.
cg at 0 forever
m girl * distance from 0 = m plank * distance from 0
center of plank moves 3 to right if girl now 2 meters from CG of system to left
3*12.6 = 2 m
m = 3 * 6.3 = 18.9 kg
To solve this problem, we can use the principle of conservation of momentum.
1. Calculate the initial momentum of the system:
- Let the mass of the girl be represented by 'm'.
- The initial velocity of the girl relative to the plank is 2 m/s.
- The mass of the plank is 12.6 kg.
- The initial velocity of the plank is 0 m/s (as it is stationary).
- The initial momentum of the girl is given by: (m)(2 m/s).
- The initial momentum of the plank is given by: (12.6 kg)(0 m/s) = 0 kg·m/s.
2. Calculate the final momentum of the system:
- Once the girl reaches the end of the plank, the center of mass of the system is 2m from the end of the plank.
- The mass of the girl remains 'm'.
- The final velocity of the girl relative to the plank is 0 m/s (since she reached the end of the plank).
- The final velocity of the plank is unknown.
- The final momentum of the girl is given by: (m)(0 m/s) = 0 kg·m/s.
- The final momentum of the plank is given by: (12.6 kg)(v) , where 'v' is the final velocity of the plank.
3. Apply the conservation of momentum:
- According to the principle of conservation of momentum, the total initial momentum of a system must be equal to the total final momentum of the system.
- So, (m)(2 m/s) + (12.6 kg)(0 m/s) = (m)(0 m/s) + (12.6 kg)(v).
- Solving for 'v', we get: v = (m)(2 m/s) / (12.6 kg).
- But we also know that the center of mass is 2m from the end of the plank, so v = 2 m/s.
4. Substitute the value of 'v' back into the equation:
- (m)(2 m/s) + (12.6 kg)(0 m/s) = (m)(0 m/s) + (12.6 kg)(2 m/s).
- Simplifying, we get: 2m + 0 = 0 + 25.2 kg·m/s.
- Canceling out the zeros, we have: 2m = 25.2 kg·m/s.
5. Solve for 'm':
- Dividing both sides of the equation by 2, we get: m = 25.2 kg·m/s / 2.
- Simplifying, we find: m = 12.6 kg.
Therefore, the mass of the girl is 12.6 kg.
To solve this problem, we need to consider the conservation of momentum and the fact that the center of mass of the system remains stationary.
Given:
Length of the plank (L) = 10 m
Velocity of the girl relative to the plank (v) = 2 m/s
Distance of the center of mass from the end of the plank (d) = 2 m
Mass of the plank (M) = 12.6 kg
We can use the equation for the center of mass of a two-object system:
M1 * d1 + M2 * d2 = M * D
Where:
M1 and M2 are the masses of the two objects (girl and plank), respectively,
d1 and d2 are the distances from the center of mass of the system to each object, respectively,
M is the total mass of the system,
D is the distance from the center of mass of the system to the end of the plank.
Let's solve for the mass of the girl (M1):
M1 * d1 + M2 * d2 = M * D
M1 * d + M2 * (L - d) = M * L
Substituting the given values:
M1 * 2 + 12.6 * (10 - 2) = (M1 + 12.6) * 10
2M1 + 10 * 12.6 - 12.6 * 2 = 10M1 + 12.6 * 10
25.2 - 25.2 + 20M1 = 126
20M1 = 126
M1 = 126 / 20
M1 = 6.3 kg
Therefore, the mass of the girl is 6.3 kg when the mass of the plank is 12.6 kg.