Question

4. (a) A system consists of particles of masses 5,2 and 3 grams located at \( (1,0,-1),(1,2,1) \) and \( (1,1,-3) \), respectively. Find the position of centre of mass of the system of particles. (b) Locate the centre of mass of a system of particles of masses \( m_{1}=1 \mathrm{~kg} ; m_{2}=2 \mathrm{~kg} \) and \( m_{3}=3 \mathrm{~kg} \), situated at the corners of an equilateral triangle of side 10 cm . 5. Calculate the centre of mass of a non-uniform rod whose mass per unit length varies as \( \rho=\rho_{o} \frac{x^{2}}{L} \), where \( \rho_{o} \) is a constant, \( L \) is the length of the rod and \( x \) is the distance of any point on the rod measured from one end. 6. Find an expression for the moment of inertia of a uniform rectangular plate of mass \( M \) and dimensions \( a \) and \( b \) about an axis through the centre of the plate and normal to the plate. 7. Find the centre of mass of a homogenous semicircular plate of radius \( a \) lying in the \( x-y \) plane as shown in figure 7.1. Fig. 7.1 8. Figure 8.1 shows an annular ring. An annular ring is a disc with a concentric hole in the centre and thus has two radii; \( R \) and \( r \). determine the moment of inertia of the annular ring about an axis through its centre and perpendicular to its face.

          4. (a) A system consists of particles of masses 5,2 and 3 grams located at \( (1,0,-1),(1,2,1) \) and \( (1,1,-3) \), respectively. Find the position of centre of mass of the system of particles. (b) Locate the centre of mass of a system of particles of masses \( m_{1}=1 \mathrm{~kg} ; m_{2}=2 \mathrm{~kg} \) and \( m_{3}=3 \mathrm{~kg} \), situated at the corners of an equilateral triangle of side 10 cm .
5. Calculate the centre of mass of a non-uniform rod whose mass per unit length varies as \( \rho=\rho_{o} \frac{x^{2}}{L} \), where \( \rho_{o} \) is a constant, \( L \) is the length of the rod and \( x \) is the distance of any point on the rod measured from one end.
6. Find an expression for the moment of inertia of a uniform rectangular plate of mass \( M \) and dimensions \( a \) and \( b \) about an axis through the centre of the plate and normal to the plate.
7. Find the centre of mass of a homogenous semicircular plate of radius \( a \) lying in the \( x-y \) plane as shown in figure 7.1.

Fig. 7.1
8. Figure 8.1 shows an annular ring. An annular ring is a disc with a concentric hole in the centre and thus has two radii; \( R \) and \( r \). determine the moment of inertia of the annular ring about an axis through its centre and perpendicular to its face.

4. (a) A system consists of particles of masses 5,2 and 3 grams located at (1,0,-1),(1,2,1) and (1,1,-3), respectively. Find the position of centre of mass of the system of particles. (b) Locate the centre of mass of a system of particles of masses m1=1 kg ; m2=2 kg and m3=3 kg, situated at the corners of an equilateral triangle of side 10 cm .
5. Calculate the centre of mass of a non-uniform rod whose mass per unit length varies as ρ=ρo(x^2)/(L), where ρo is a constant, L is the length of the rod and x is the distance of any point on the rod measured from one end.
6. Find an expression for the moment of inertia of a uniform rectangular plate of mass M and dimensions a and b about an axis through the centre of the plate and normal to the plate.
7. Find the centre of mass of a homogenous semicircular plate of radius a lying in the x-y plane as shown in figure 7.1.

Fig. 7.1
8. Figure 8.1 shows an annular ring. An annular ring is a disc with a concentric hole in the centre and thus has two radii; R and r. determine the moment of inertia of the annular ring about an axis through its centre and perpendicular to its face.

Added by John D.

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