The vector position of a 3.65 g particle moving in the xy plane varies in time according to the following equation.
r1 = <(3i+3j)t + 2jt^2>
At the same time, the vector position of a 5.60 g particle varies according to the following equation.
r2 = <3i - 2it^2 - 6jt>
For each equation, t is in s and r is in cm. Solve the following when t = 2.90
(a) Find the vector position of the center of mass.
i-hat__________ cm
j-hat___________cm
(b) Find the linear momentum of the system.
i-hat___________ g-cm/s
j-hat____________g-cm/s
(c) Find the velocity of the center of mass.
i-hat_________cm/s
j-hat__________ cm/s
(d) Find the acceleration of the center of mass.
i-hat____________cm/s2
j-hat_____________cm/s2
(e) Find the net force exerted on the two-particle system.
i-hat___________ μN
j-hat___________μN
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Maybe someone could just explain to me how to solve for each part because I'm thoroughly confused. THANK YOU.
To solve each part of the problem, we can follow these steps:
Step 1: Substitute the given values of t into the equations for r1 and r2.
Step 2: Calculate the vector position of the center of mass (r_cm) by using the formula:
r_cm = (m1 * r1 + m2 * r2) / (m1 + m2)
where m1 and m2 are the masses of the particles.
Step 3: Calculate the linear momentum of the system (p_sys) by using the formula:
p_sys = m1 * v1 + m2 * v2
where v1 and v2 are the velocities of the particles, which can be found by taking the derivatives of r1 and r2 with respect to time.
Step 4: Calculate the velocity of the center of mass (v_cm) using the formula:
v_cm = p_sys / (m1 + m2)
Step 5: Calculate the acceleration of the center of mass (a_cm) by taking the derivative of v_cm with respect to time.
Step 6: Finally, calculate the net force exerted on the two-particle system using Newton's second law:
F_net = (m1 + m2) * a_cm
Let's now go through each part one by one.
(a) Find the vector position of the center of mass:
Substitute t = 2.90 into the equations for r1 and r2:
r1(2.90) = <(3i+3j)(2.90) + 2j(2.90)^2>
r2(2.90) = <3i - 2i(2.90)^2 - 6j(2.90)>
Next, calculate r_cm:
r_cm = (m1 * r1(2.90) + m2 * r2(2.90)) / (m1 + m2)
Substitute the masses of the particles into this equation and calculate the result.
(b) Find the linear momentum of the system:
To find the velocities of the particles (v1 and v2), take the derivatives of r1 and r2 with respect to time and substitute t = 2.90. Then calculate p_sys using the formula mentioned above.
(c) Find the velocity of the center of mass:
Calculate v_cm using the formula mentioned above and substitute the values of p_sys, m1, and m2.
(d) Find the acceleration of the center of mass:
Calculate a_cm by taking the derivative of v_cm with respect to time. Substitute the values of m1, m2, and the derivative of p_sys with respect to time.
(e) Find the net force exerted on the two-particle system:
Calculate F_net using Newton's second law by substituting the values of m1, m2, and a_cm into the equation.
Remember to use consistent units throughout the calculations.