Answer:
![w_2=0.467rad/s](https://tex.z-dn.net/?f=w_2%3D0.467rad%2Fs)
Explanation:
Four people standing on the ground each of mass and usually this questions have to find the final angular velocity
![m_t=4*70kg=280kg](https://tex.z-dn.net/?f=m_t%3D4%2A70kg%3D280kg)
The radius ![r=4.2/2=2.1m](https://tex.z-dn.net/?f=r%3D4.2%2F2%3D2.1m)
Angular velocity ![w_1=0.79rad/s](https://tex.z-dn.net/?f=w_1%3D0.79rad%2Fs)
The moment of inertia total is ![I_t=1790 kg/m^2](https://tex.z-dn.net/?f=I_t%3D1790%20kg%2Fm%5E2)
Momento if inertia
![I_1=m_t*r^2](https://tex.z-dn.net/?f=I_1%3Dm_t%2Ar%5E2)
![I_1=280kg*(2.1m)^2=1234.8kg*m^2](https://tex.z-dn.net/?f=I_1%3D280kg%2A%282.1m%29%5E2%3D1234.8kg%2Am%5E2)
Angular momentum
![I_1*w_1=I_t*w_2](https://tex.z-dn.net/?f=I_1%2Aw_1%3DI_t%2Aw_2)
Solve to w2
![w_2=\frac{I_1*w_1}{I_t}](https://tex.z-dn.net/?f=w_2%3D%5Cfrac%7BI_1%2Aw_1%7D%7BI_t%7D)
![w_2=\frac{1790kg*m^2*0.79rad/s}{3024.8kg*m^2}](https://tex.z-dn.net/?f=w_2%3D%5Cfrac%7B1790kg%2Am%5E2%2A0.79rad%2Fs%7D%7B3024.8kg%2Am%5E2%7D)
![w_2=0.467rad/s](https://tex.z-dn.net/?f=w_2%3D0.467rad%2Fs)
Answer:
126.99115 g
Explanation:
50 g at 90 cm
Stick balances at 61.3 cm
x = Distance of the third 0.6 kg mass
Meter stick hanging at 50 cm
Torque about the support point is given by (torque is conserved)
![mgl_1=Mgl_2\\\Rightarrow M=\dfrac{ml_1}{l_2}\\\Rightarrow M=\dfrac{50\times (61.3-90)}{50-61.3}\\\Rightarrow M=126.99115\ g](https://tex.z-dn.net/?f=mgl_1%3DMgl_2%5C%5C%5CRightarrow%20M%3D%5Cdfrac%7Bml_1%7D%7Bl_2%7D%5C%5C%5CRightarrow%20M%3D%5Cdfrac%7B50%5Ctimes%20%2861.3-90%29%7D%7B50-61.3%7D%5C%5C%5CRightarrow%20M%3D126.99115%5C%20g)
The mass of the meter stick is 126.99115 g
Answer with explanation:
We are given that
Mass of ball,
75 g=![\frac{75}{1000}=0075kg](https://tex.z-dn.net/?f=%5Cfrac%7B75%7D%7B1000%7D%3D0075kg)
1 kg=1000 g
Height,![h_1=1.6 m](https://tex.z-dn.net/?f=h_1%3D1.6%20m)
![h_2=0.6 m](https://tex.z-dn.net/?f=h_2%3D0.6%20m)
Horizontal velocity,![v_x=2 m/s](https://tex.z-dn.net/?f=v_x%3D2%20m%2Fs)
Mass of plate![m_2=400 g=\frac{400}{1000}=0.4 kg](https://tex.z-dn.net/?f=m_2%3D400%20g%3D%5Cfrac%7B400%7D%7B1000%7D%3D0.4%20kg)
a.Initial velocity of plate,![u_2=0](https://tex.z-dn.net/?f=u_2%3D0)
Velocity before impact=![u_1=\sqrt{2gh_1}=\sqrt{2\times 9.8\times 1.6}=5.6m/s](https://tex.z-dn.net/?f=u_1%3D%5Csqrt%7B2gh_1%7D%3D%5Csqrt%7B2%5Ctimes%209.8%5Ctimes%201.6%7D%3D5.6m%2Fs)
Where ![g=9.8 m/s^2](https://tex.z-dn.net/?f=g%3D9.8%20m%2Fs%5E2)
Velocity after impact,![v_1=\sqrt{2gh_2}=\sqrt{2\times 9.8\times 0.6}=3.4m/s](https://tex.z-dn.net/?f=v_1%3D%5Csqrt%7B2gh_2%7D%3D%5Csqrt%7B2%5Ctimes%209.8%5Ctimes%200.6%7D%3D3.4m%2Fs)
According to law of conservation of momentum
![m_1u_1+m_2u_1=-m_1v_1+m_2v_2](https://tex.z-dn.net/?f=m_1u_1%2Bm_2u_1%3D-m_1v_1%2Bm_2v_2)
Substitute the values
![0.075\times 5.6+0=-0.075\times 3.4+0.4v_2](https://tex.z-dn.net/?f=0.075%5Ctimes%205.6%2B0%3D-0.075%5Ctimes%203.4%2B0.4v_2)
![0.4v_2=0.075\times 5.6+0.075\times 3.4](https://tex.z-dn.net/?f=0.4v_2%3D0.075%5Ctimes%205.6%2B0.075%5Ctimes%203.4)
![v_2=\frac{0.075\times 5.6+0.075\times 3.4}{0.4}=1.69 m/s](https://tex.z-dn.net/?f=v_2%3D%5Cfrac%7B0.075%5Ctimes%205.6%2B0.075%5Ctimes%203.4%7D%7B0.4%7D%3D1.69%20m%2Fs)
Velocity of plate=1.69 m/s
b.Initial energy=![\frac{1}{2}m_1v^2_x+m_1gh_1=\frac{1}{2}(0.075)(2^2)+0.075\times 9.8\times 1.6=1.326 J](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7Dm_1v%5E2_x%2Bm_1gh_1%3D%5Cfrac%7B1%7D%7B2%7D%280.075%29%282%5E2%29%2B0.075%5Ctimes%209.8%5Ctimes%201.6%3D1.326%20J)
Final energy=![\frac{1}{2}m_1v^2_x+m_1gh_2+\frac{1}{2}m_2v^2_2](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7Dm_1v%5E2_x%2Bm_1gh_2%2B%5Cfrac%7B1%7D%7B2%7Dm_2v%5E2_2)
Final energy=![\frac{1}{2}(0.075)(2^2)+0.075\times 9.8\times 0.6+\frac{1}{2}(0.4)(1.69)^2=1.162 J](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%280.075%29%282%5E2%29%2B0.075%5Ctimes%209.8%5Ctimes%200.6%2B%5Cfrac%7B1%7D%7B2%7D%280.4%29%281.69%29%5E2%3D1.162%20J)
Energy lost due to compact=Initial energy-final energy=1.326-1.162=0.164 J