Answer :
Answer:
a) P_total = (4.1 i -1.7 j + 5k) kg m/s
b) F_g = 12.753 N
c) P_total = (4.1 i -2.46518 j + 5k) kg m/s
Explanation:
Given:
- The mass of ball 1, m_1 = 0.5 kg
- The mass of the ball 2, m_2 = 0.8 kg
- The velocity of ball 1, V_1 = (5 i -5 j + 2k)
- The velocity of ball 2, V_2 = (2 i +1 j + 5k)
Find:
(a) At this Instant, what is the total momentum of the device?
(b) What is the net gravitational (vector) force exerted by the Earth on the device?
(c) At a time 0.06 seconds later, what is the total momentum of the device?
Solution:
- We will take unit vectors i , j , k in the directions horizontal, vertical, and out of the page respectively.
- The total momentum of the system is the sum of momentum of individual objects in a system:
P_total = P_1 + P_2
- Where, P_1 : Momentum for ball 1. P_2 : Momentum for ball 2.
- The momentum of an object is the scalar multiple of the velocity vector and its mass. Hence, we will compute the total momentum as follows:
P_total = m_1*V_1 + m_2*V_2
P_total = 0.5*(5 i -5 j + 2k) + 0.8*(2 i +1 j + 5k)
Hence, the total momentum is:
P_total = (4.1 i -1.7 j + 5k) kgm/s
- The net gravitational Force exerted by the earth on the device is due to the weight of each mass as follows:
F_g = W_1 + W_2
- Where, W_1 : Weight for ball 1. W_2 : Weight for ball 2.
- Hence, the net gravitational force is as follows:
F_g = m_1*g + m_2*g = g*(m_1 + m_2)
F_g = 9.81*(0.8 + 0.5)
F_g = 12.753 N
- The momentum of the system after a certain time under the experience of gravitational force will affect the initial velocity of the balls in the system. So to calculate the new velocities of the ball. we will apply kinematic equation of motion on j vector of the both balls, in which gravitational acceleration acts.
- V'_1 = (5 i - (5 + g*t) j + 2k)
V'_2 = (2 i +(1 - g*t) j + 5k)
- Where, V'_1 and V'_2 are new velocities of the ball. Hence, we compute:
V'_1 = (5 i - (5 + 9.81*0.06) j + 2k)
V'_2 = (2 i +(1 - 9.81*0.06) j + 5k)
- Hence, two velocities are:
V'_1 = (5 i - 5.5886 j + 2k)
V'_2 = (2 i + 0.4114 j + 5k)
- The momentum of an object is the scalar multiple of the velocity vector and its mass. Hence, we will compute the total momentum as follows:
P_total = m_1*V'_1 + m_2*V'_2
P_total = 0.5*(5 i -5.5886 j + 2k) + 0.8*(2 i +0.4114 j + 5k)
Hence, the total momentum is:
P_total = (4.1 i -2.46518 j + 5k) kgm/s
The total momentum of the balls is [tex](6.2 j - 2.2j+ 7.1 k) \ kgm/s[/tex].
The net gravitational force exerted by the Earth on the device is 12.74 N.
The total momentum of the system after 0.06s is 0.764 kgm/s.
The given parameters:
- Velocity of the 0.5 kg ball, (6, -6, 3) m/s
- Velocity of the 0.8 kg ball, (4, 1, 7), m/s
Total momentum of the balls
The total momentum of the balls is calculated by applying the principle of conservation of linear momentum;
[tex]P = m_1 u_1 + m_2 u_2\\\\P = 0.5(6, -6, 3) + 0.8(4, 1, 7)\\\\P = (3i-3j+ 1.5k) + (3.2i + 0.8j + 5.6k)\\\\P = (6.2 j - 2.2j+ 7.1 k) \ kgm/s[/tex]
The net gravitational force
[tex]F = m_1 g + m_2 g\\\\F = g(m_1 +m_2)\\\\F = 9.8(0.5 + 0.8)\\\\F = 12.74 \ N[/tex]
The total momentum of the system after 0.06s
[tex]P = Ft\\\\P = 12.74 \times 0.06\\\\P = 0.764 \ kgm/s[/tex]
The missing part of the question is below;
At a particular instant, the 0.5 kg ball has a velocity (6, -6, 3) m/s and the 0.8 kg ball has a velocity (4, 1, 7) m/s.
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