The vector position of a 3.45 g particle moving in the xy plane varies in time according to
1 = 3 + 3t + 2t2
where t is in seconds and is in centimeters. At the same time, the vector position of a 5.00 g particle varies as 2 = 3 − 2t2 − 6t.
Determine the vector position of the center of mass at t = 2.80.
To determine the vector position of the center of mass at t = 2.80, we need to find the individual positions of the two particles and then calculate the center of mass using their respective masses.
Let's start by finding the position vector for the first particle. The position vector for the first particle is given by:
r₁(t) = x₁(t)i + y₁(t)j
Where x₁(t) and y₁(t) represent the x and y components of the position vector respectively.
Given that:
x₁(t) = 3 + 3t + 2t²
To find the x component of the position vector at t = 2.80, substitute t = 2.80 into the equation:
x₁(2.80) = 3 + 3(2.80) + 2(2.80)²
x₁(2.80) = 3 + 8.4 + 15.68
x₁(2.80) = 27.08
Similarly, we can find the y component of the position vector for the first particle using the given equation:
y₁(t) = 1
Since y₁(t) is constant, the y component of the position vector remains the same regardless of the time. Therefore, y₁(t) = 1 at any value of t.
Now, let's find the position vector for the second particle. The position vector for the second particle is given by:
r₂(t) = x₂(t)i + y₂(t)j
Where x₂(t) and y₂(t) represent the x and y components of the position vector respectively.
Given that:
x₂(t) = 3 - 2t² - 6t
To find the x component of the position vector at t = 2.80, substitute t = 2.80 into the equation:
x₂(2.80) = 3 - 2(2.80)² - 6(2.80)
x₂(2.80) = 3 - 2(7.84) - 16.8
x₂(2.80) = -29.68
Similarly, we can find the y component of the position vector for the second particle using the given equation:
y₂(t) = 2
Since y₂(t) is constant, the y component of the position vector remains the same regardless of the time. Therefore, y₂(t) = 2 at any value of t.
Now that we have the x and y components of both position vectors, we can calculate the position vector of the center of mass using the formula:
R_cm = (m₁r₁ + m₂r₂) / (m₁ + m₂)
Where R_cm is the position vector of the center of mass, m₁ and m₂ are the masses of the first and second particles, and r₁ and r₂ are the position vectors of the first and second particles respectively.
Given that m₁ = 3.45 g and m₂ = 5.00 g, we need to convert these masses to kilograms before proceeding with the calculation.
m₁ = 3.45 g / 1000 = 0.00345 kg
m₂ = 5.00 g / 1000 = 0.00500 kg
Substituting the values, we get:
R_cm = (0.00345 * [27.08i + 1j] + 0.00500 * [-29.68i + 2j]) / (0.00345 + 0.00500)
Simplifying the expression, we get:
R_cm = (0.09366i + 0.00345j - 0.1476i + 0.01000j) / 0.00845
R_cm = (0.09366i - 0.1476i + 0.00345j + 0.01000j) / 0.00845
R_cm = (0.09366i - 0.1476i + 0.00345j + 0.01000j) / 0.00845
R_cm = -0.0549i + 1.0648j
Therefore, the vector position of the center of mass at t = 2.80 is approximately -0.0549i + 1.0648j.