Sheet 03 Force law for S.H.M. and Expression of Time period.
A mass m attached to a spring oscillates with a period of \(2s\). If the mass is increased by \(3Kg\), the period increases by \(1s\). find; the initial mass, assuming that Hooke’s law is obeyed.
Ans: \(m=2.4Kg\).
A pan of mass \(1Kg\) mass is attached to a spring balance. A weight of \(2Kg\) when placed on the pan stretches the spring by \(10cm\). What is the frequency with which the empty pan will oscillates?
Ans: \(\nu=\frac{7}{\pi}Hz\).
A small trolley of mass \(2.0Kg\) resting on a horizontal turn table is connected by a light spring to the center of the table. When the turn table is set into rotation a speed of \(600rpm\), the length of the spring is \(50cm\). If, the original length of the spring is \(42cm\), find force constant of the spring.
Ans: \(K=49387.7\frac{N}{m}\).
A \(200g\) of mass hangs at the end of a spring. When \(20g\) more mass is added to the end of the spring, it stretches \(7cm\) more. If, \(20g\) mass is removed. What will be the period of vibration of the system?
Ans: \(T=1.66s\).
A uniform spring whose un-stretched length is \(l\) has a force constant \(K\). The spring is cut in to two pieces of un-stretched lengths \(l_{1}\) and \(l_{2}\). Where \(l_{1}=nl_{2}\) and \(n\) is an integer. What are the corresponding force constants \(K_{1}\) and \(K_{2}\) in terms of \(n\) and \(K\).
Ans: \(K_{1}=\left(\frac{n-1}{n}\right)K\) and \(K_{2}=\left(n+1\right)K\).
A body of mass \(12Kg\) is suspended by a coil spring of natural length \(50cm\) and force constant \(\left(2.0\times10^{3}\right)\frac{N}{m}\). What is the stretched length of spring? If the body is pulled down further stretching the spring to a length of \(59cm\) and then released, what is the frequency of oscillations of the suspended mass? (neglect the mass of the spring).
Ans: \(\nu=2.05s^{-1}\).
Two springs are joined and are connected to a mass \(m\) as shown in figure. If the spring force constants are \(K_{1}\) and \(K_{2}\), then show thatthe frequency of the oscillations of mass \(m\) is given by \(\nu=\frac{1}{2\pi}\sqrt{\frac{K_{1}K_{2}}{(K_{1}+K_{2})m}}\).
Two masses \(m_{1}=1.0Kg\) and \(m_{2}=0.5Kg\) are suspended together by a massless spring of force constant \(K=12.5\frac{N}{m}\). When they are in equilibrium position, \(m_{1}\) is gently removed. Calculate: the angular frequency and amplitude of oscillations of \(m_{2}\). Given: \(g=10\frac{m}{s^{2}}\).
Ans: \(\omega=5\frac{rad}{sec}\) and \(r=0.8m\).
A trolley of mass \(6.0kg\) is connected to the two identical springs each of force constant \(600\frac{N}{m}\), as shown in the figure. If, trolley is displaced from its equilibrium position by \(5cm\) and released, (I) what is the period of oscillations? (II) The maximum speed of the trolley? (III) How much is the total energy dissipated as heat by the time trolley comes to rest due to damping force?
Ans:(I) \(T=4.44s\), (II) \(V_{max}=0.707\frac{m}{s}\) and (III) \(E=3.0J\).
A tray of mass \(12kg\) is suspended by two identical springs as shown in below figure. When the tray is pressed down slightly and released, it executes S.H.M with a time period of \(1.5s\). What is the force constant of each spring? When a block of mass \(M\) is placed on the tray the period of oscillations changes to \(3.0s\). What is the mass of the block?
Ans: \(K=(1.053\times10^{2})\frac{N}{m}\) and \(M=36kg\).
Two identical springs, each of spring factor \(K\) may be connected in following ways as shown in below figure; Deduce the spring factor for the oscillations of the body \(P\) in each case.
Ans:(a) \(K_{1}=2k\) (b) \(K_{2}=\frac{K}{2}\) and (c) \(K_{3}=2K\).
An impulsive force gives an initial velocity of \(-1.0\frac{m}{s}\) to the mass in the un-stretched spring position as shown below. What is the amplitude of the motion? Give \(Y\) as the function of time \(t\) for oscillating mass. Given: \(m(mass)=3.0kg\) and \(K(spring-Constant)=1200\frac{N}{m}\).
Ans: \(r=5cm\) and \(Y=-5Sin20t\).
A \(5kg\) collar is attached to a spring of force constant \(500\frac{N}{m}\). It slides without friction on a horizontal rod as shown in figure. The collar is displaced form its equilibrium position by \(10.0cm\) and released. Calculate: (i) the time period of oscillations? (ii) The maximum speed and (iii) the maximum acceleration of the collar.
Ans:(i) \(T=0.628s\), (ii) \(V_{max}=1.0\frac{m}{s}\) and (iii) \(a_{max}=10\frac{m}{s^{2}}\).
