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Answer :
Final answer:
The angular momentum of a particle about the origin is given by the cross product of its position vector and linear momentum. In this case, the position vector of the particle is (6.00 î + 5.70 t ĵ) and the mass of the particle is 1.70 kg. To find the angular momentum, calculate the linear momentum and take the cross product with the position vector.
Explanation:
The angular momentum of a particle about the origin is given by the cross product of its position vector and linear momentum. In this case, the position vector of the particle is represented by r = (6.00 î + 5.70 t ĵ) and the mass of the particle is 1.70 kg. To find the angular momentum, we need to calculate the linear momentum first.
The linear momentum of a particle is given by the product of its mass and velocity. The velocity vector is given by the derivative of the position vector with respect to time, which in this case is v = (0 î + 5.70 ĵ) m/s. Substituting the values, we can find the linear momentum, which is p = (1.70 kg)(0 î + 5.70 ĵ) m/s.
To find the angular momentum, we take the cross product of the position vector and linear momentum:
Å• × p = (6.00 î + 5.70 t ĵ) × (1.70 kg)(0 î + 5.70 ĵ) m/s = (0 î + 34.65 î t + 9.69 ĵ) kg·m²/s
Therefore, the angular momentum of the particle about the origin as a function of time is (0 î + 34.65 î t + 9.69 ĵ) kg·m²/s.
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Rewritten by : Jeany
The angular momentum of the particle about the origin as a function of time is L(t) = 58.14 t k kg·m²/s. This result was obtained by calculating the cross product of the position and linear momentum vectors.
To determine the angular momentum of a particle about the origin as a function of time, we start by using the given position vector r(t) = (6.00 î + 5.70t ĵ) in meters.
The linear momentum p of the particle is given by p = m v, where m is the mass and v is the velocity. Since mass m = 1.70 kg, we first need the velocity. The velocity v(t) is the derivative of the position vector:
v(t) = d(r(t))/dt = 0 î + 5.70 ĵ = 5.70 ĵ m/s
Now, the linear momentum p(t) is:
p(t) = m v(t) = 1.70 kg * 5.70 ĵ m/s = 9.69 ĵ kg·m/s
The angular momentum L about the origin is given by L = r × p. Performing the cross product calculation:
r = 6.00 î + 5.70t ĵ
p = 9.69 ĵ
r × p = (6.00 î + 5.70t ĵ) × (9.69 ĵ)
Calculating the cross product, we get:
- i-component: 0
- j-component: 0
- k-component: 6.00 * 9.69 - 0 = 58.14 t k
Thus, the angular momentum as a function of time is:
L(t) = 58.14 t k kg·m²/s
