A cantilever beam OP is connected to another beam PQ with a pin joint as shown in figure. A load of $$10$$ kN is applied at the mid-point of PQ. The magnitude of bending moment (in kN-m) at fixed end $$O$$ is

The deflection of a spring with 20 active turns under a load of 1000 N is 10 mm. The spring is made into two pieces each of 10 active coils and placed in parallel under the same load. The deflection of the system is

A hollow shaft (d₀ = $$2$$d_i, where d₀ and d_i are the outer and inner diameters respectively) needs to transmit $$20$$ kW power at $$3000$$ RMP. If the maximum permissible shear stress is $$30$$MPa, d₀ is

A rope brake dynamometer attached to the crank shaft of an $$I.C.$$ engine measures a brake power of $$10$$ kW when the speed of rotation of the shaft is $$400$$ rad/s. The shaft torque (in N-m) sensed by the dynamometer is ________