A big olive (m = 0.40 kg) lies at the origin of an xy coordinate system, and a big Brazil nut (M = 1.2 kg) lies at the point (1.0,2.0) m. At t = 0, a force o = (2.0 + 3.0) N begins to act on the olive, and a force n = (-3.0 - 2.0) N begins to act on the nut. In unit-vector notation, what is the displacement of the center of mass of the olive-nut system at t = 5.0 s, with respect to its position at t = 0?
To calculate the displacement of the center of mass of the olive-nut system at t = 5.0 s, we need to consider the individual displacements of the olive and the nut.
1. Calculate the individual displacements of the olive and the nut.
- For the olive, we can use the equation of motion: s = ut + (1/2)at^2
- Initial velocity (u) = 0 (as it starts from rest)
- Acceleration (a) = ΣF / m, where ΣF is the sum of the forces acting on the olive, and m is the mass of the olive.
- ΣF = 2.0 N + 3.0 N = 5.0 N (sum of forces acting on the olive)
- Substituting the values, we get: s_olive = 0 * 5.0 + (1/2) * (5.0 / 0.40) * (5.0)^2 = 31.25 m
- For the nut, we can use the same equation of motion:
- Initial velocity (u) = 0 (as it starts from rest)
- Acceleration (a) = ΣF / M, where ΣF is the sum of the forces acting on the nut, and M is the mass of the nut.
- ΣF = -3.0 N - 2.0 N = -5.0 N (sum of forces acting on the nut)
- Substituting the values, we get: s_nut = 0 * 5.0 + (1/2) * (-5.0 / 1.20) * (5.0)^2 = -52.08 m
2. Calculate the displacement of the center of mass:
- The center of mass is calculated using the formula:
r_cm = (m_olive * r_olive + m_nut * r_nut) / (m_olive + m_nut)
- Here, m_olive and m_nut are the masses of the olive and the nut, respectively.
- r_olive is the displacement of the olive.
- r_nut is the displacement of the nut.
- Substituting the values, we get:
r_cm = (0.40 kg * 31.25 m + 1.2 kg * (-52.08 m)) / (0.40 kg + 1.2 kg)
= (-34.875 + (-62.496)) / 1.6
= -97.371 / 1.6
= -60.8575 m (rounded to four decimal places)
3. Express the result in unit vector notation:
- Since the origin is the reference point, the displacement can be expressed as a vector from the origin to the center of mass point.
- The x-component of the displacement is -60.8575 m and the y-component is 0 m.
- Thus, the displacement in unit vector notation is r_cm = (-60.8575i + 0j) m, where i and j represent the unit vectors in the x and y directions, respectively.
Therefore, the displacement of the center of mass of the olive-nut system at t = 5.0 s, with respect to its position at t = 0, is given by (-60.8575i + 0j) m.