P. 4118. The plank of mass 2m shown in the figure can move without friction along a horizontal surface. The length of the thread of the pendulum fixed to the plank is L=20 cm, and the mass of the (point-like) bob at the end of the thread is m. The bob is displaced such that the angle between the thread and the vertical is =60o, and then it is released.
a) Find the velocity of the bob when the thread is vertical.
b) Find the velocity of the plank when the angle between the thread and the vertical is 45o.
P. 4121. A piston, which can move without friction, closes a sample of Helium in a vertical, thermally insulated cylinder, which is closed at its bottom, and the cross-section of which is A=2 dm2. Above the piston there is a fixed stand to which an unstretched spring of spring constant D=2000 N/m is attached, whose bottom end is at a distance of d=2 dm from the piston when the volume of the gas is V0=8 dm3. The external pressure is p0=105 Pa, g=10 m/s2. The gas confined in the cylinder is heated with an electric heating element which is automatically controlled such that the piston is continuously moving with a constant speed of v0=5 cm/s.
a) Determine and plot the instantaneous power of the heating element as a function of the time, while the volume of the gas increases from V0 to V1=18 dm3.
b) Determine the heat transferred by the electric heating element to the expanding gas in the process.