PHYS ILL (ii) Write down the expressions for the tension T_(1) in the left string in terms of unit vectors I and D and the angle that the string makes with the x-axis. (iii) Write down the expression for the tension T_(z) in the right string in terms of unit vectors I and I and the angle that the string makes with the x-axis. (1 mark) (1 mark) (w) Then using the expressions above determine an expression for the total net force acting on the mass given that the weight of the mass is given by overrightarrow (w)=mg(-3) (v) Then using the above result in conjunction with the condition for static equilibrium calculate the tension T in the rope. (3 marks) (2 marks) OUESTION 4:00 MARKS) mass is placed at the top of a smooth frictionless inclined plane of 4m length which makes an ungle of 30^circ with the horizontal and slides down the plane. Using the principle of conservation of energy calculate the speed on the mass when it reaches the boltum of the plane. (5 marks) (b) A. bomb of music duku explodes in flight at the instant its velocity is overrightarrow (a)=(10i+5j)ms^-1 . It breaks into two pieces m. with mass 10 kis and wash with muss 30kg Mass in, has a velocity overrightarrow (V)_(1) and my has a velocity delta _(2)=(100i+50j+10k)ms^-1 A.suming that the conserved. calculate the velocity of the music Finis (5 marks) first particle I is at the origin, the second computer trình C. 100 while she third particle 3 is at (sqrt (2),y) Determine the value of y for the third particle. (2 marks) (b) Calculate the position vector of the centre of musi of the collection of particles (c) If a fourth particle of mass m is added to die collection of particles, then where should this purticle be placed so that the centre of mass is located at the origin? (5 marks) (3 marks)

(ii) The tension \( T_1 \) in the left string can be expressed in terms of unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) and the angle \( \theta \) that the string makes with the x-axis as:<br />\[ T_1 = T_1 \cos(\theta) \mathbf{i} + T_1 \sin(\theta) \mathbf{j} \]<br /><br />(iii) The tension \( T_2 \) in the right string can be expressed in terms of unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) and the angle \( \phi \) that the string makes with the x-axis as:<br />\[ T_2 = T_2 \cos(\phi) \mathbf{i} + T_2 \sin(\phi) \mathbf{j} \]<br /><br />(iv) The total net force acting on the mass is given by the sum of the components of the tensions in the strings and the weight of the mass:<br />\[ \mathbf{F}_{\text{net}} = T_1 \cos(\theta) \mathbf{i} + T_1 \sin(\theta) \mathbf{j} + T_2 \cos(\phi) \mathbf{i} + T_2 \sin(\phi) \mathbf{j} + mg(-3) \mathbf{j} \]<br /><br />(v) For static equilibrium, the total net force must be zero. Therefore, the tension \( T \) in the rope can be calculated by setting the sum of the forces to zero and solving for \( T \):<br />\[ T_1 \cos(\theta) + T_2 \cos(\phi) + mg(-3) = 0 \]<br />\[ T_1 \sin(\theta) + T_2 \sin(\phi) = 0 \]<br /><br />QUESTION 4: Using the principle of conservation of energy, the speed of the mass when it reaches the bottom of the plane can be calculated as:<br />\[ \frac{1}{2}mv^2 = mgh \]<br />\[ v = \sqrt{2gh} \]<br />where \( g \) is the acceleration due to gravity and \( h \) is the height of the plane.<br /><br />(b) Using the conservation of momentum, the velocity of the music fragment \( \mathbf{V}_1 \) can be calculated as:<br />\[ m\mathbf{a} = m\mathbf{V}_1 + 30\mathbf{V}_2 \]<br />\[ \mathbf{V}_1 = \mathbf{a} - 3\mathbf{V}_2 \]<br /><br />(c) The value of \( y \) for the third particle can be determined using the given coordinates of the first and third particles.<br /><br />(d) The position vector of the center of mass of the collection of particles can be calculated as:<br />\[ \mathbf{R}_{\text{cm}} = \frac{\sum m_i \mathbf{r}_i}{\sum m_i} \]<br />where \( m_i \) is the mass of the \( i \)-th particle and \( \mathbf{r}_i \) is the position vector of the \( i \)-th particle.<br /><br />(e) If a fourth particle of mass \( m \) is added to the collection of particles, the center of mass should be located at the origin. Therefore, the position vector of the fourth particle should be:<br />\[ \mathbf{r}_4 = -\frac{\sum m_i \mathbf{r}_i}{m} \]