TOPIC E: OSCILLATIONS EXAMPLES SPRING Q1. Find general solutions for the following differential equations:

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1 TOPIC E: OSCILLATIONS EXAMPLES SPRING 2019 Mathematics of Oscillating Systems Q1. Find general solutions for the following differential equations: Undamped Free Vibration Q2. A 4 g mass is suspended by a spring of stiffness = 900 N m 1. Find: the static displacement at equilibrium; the natural circular frequency of oscillation. 4 g = 900 N/m = 100 N/m = 300 N/m Q3. A bloc of mass 16 g is suspended vertically by two light springs of stiffness 100 N m 1 and 300 N m 1 hung end-to-end as shown. Find: the stiffness of an equivalent single spring; the period of oscillation. 16 g Q4. m = 600 N/m In the system shown a mass m sits on a cart of mass 35 g 35 g which is tethered to a fixed wall by a spring of stiffness = 600 N m 1. The cart moves on frictionless rollers. Assuming that it does not slip, what mass m must be placed on the top of the cart in order that the period of oscillation be 2 s? What is the minimum coefficient of friction μ such that the mass will not slip if the cart is pulled 0.3 m away from the equilibrium position and then released? If the coefficient of friction were actually half that in part, what would be the initial acceleration of: (i) the cart; (ii) the added mass m. m m Q5. Calculate the extension at equilibrium and the natural circular frequency ω of the system shown. Friction forces and the masses of the cables and pulleys are negligible. Mechanics Examples for Topic E (Oscillations) - 1 David Apsley

2 m Q6. Find the period of oscillation of the system shown. Neglect friction and the mass of the pulley. Q7. (Exam 2010) Two objects A and B, of masses 4 g and 2 g respectively, are connected by a light inextensible cord passing over a smooth bar (see the figure). B is connected to the ground by a light spring of stiffness 200 N m 1 and unstretched length 0.5 m. You may assume that the cord connecting the objects remains taut. Find the extension of the spring when the system is in equilibrium. A 4 g 2 g B The system is released from rest from the position in which A and B are both 0.5 m from the ground. Find the distance that A descends before coming instantaneously to rest. Find the speed of A as it passes through the equilibrium position. (d) Show that the subsequent motion is simple harmonic, and calculate the period of oscillation. Q8. (Exam 2015) An object of mass 6 g is suspended from a ceiling by a spring of stiffness 300 N m 1 and unstretched length 0.8 m. (d) Find the equilibrium extension of the spring. Demonstrate that the mass undergoes simple harmonic motion about the position of equilibrium and find the period of oscillation. The mass is pulled down so that it is 1.3 m below the ceiling and is then released from rest. Find the maximum speed and maximum magnitude of acceleration in the subsequent motion. State, without any mathematical detail, what qualitative effect (if any) a small amount of velocity-dependent damping will have on: (i) the equilibrium extension; (ii) the period of oscillation; (iii) the amplitude of oscillation. Mechanics Examples for Topic E (Oscillations) - 2 David Apsley

3 Oscillating Systems Other than Mass-Spring Q9. A floating buoy consists of a cylinder of mass 3 g and radius 0.1 m. If it is pushed down (but not submerged) and then released, show that it undergoes SHM and find the natural circular frequency of oscillation. (The density of seawater, ρ w, is approximately 1025 g m 3 ; neglect the added mass of the seawater set in motion and assume that the buoy is stable). 0.1m y If the buoy is replaced by an inverted cone of mass m and semi-vertex angle α, write down an equation of motion in terms of submerged depth and show that any oscillatory motion is not SHM. Q10. Two small balls of mass 0.2 g are attached to the ends of a uniform rod of mass 1.2 g and length 0.5 m. The whole assembly is suspended from its centre by a torsion cable which provides a restoring torque of 0.2 N m per radian twist. Find: the period of oscillation; the energy of the system if the maximum twist is π/6 radians. Mechanics Examples for Topic E (Oscillations) - 3 David Apsley

4 Rotational Oscillations Q11. A uniform rod of mass 2 g and length 0.8 m supports a concentrated mass of 3 g at one end and is pivoted at the other. It is restrained by two springs, each of stiffness 600 N m 1, positioned on opposite sides of the rod at distances 0.4 m and 0.6 m from the pivot as shown. The springs are unstretched when the rod is horizontal and the angular displacement θ is measured from this position. axis 0.4m 0.2m 0.2m 3g (d) Find the moment of inertia of the rod-mass assembly about the axis. For an anticlocwise angular displacement θ from the horizontal, using the smallangle approximation write down expressions for: (i) the deflection of each spring (+ve for extension, ve for compression); (ii) the total net torque imposed by gravity and the two springs about the pivot. Find the equilibrium angular displacement. Find the period of small oscillations. 0.3 m 0.3 m 5 g 0.9 m 10 g Q12. A uniform rod of mass 5 g and length 0.9 m supports a concentrated mass of 10 g at one end and is pivoted at the other. It is supported by two springs, each of stiffness = 2000 N m 1, positioned as shown. Its equilibrium position is horizontal. Calculate the period of small oscillations about this equilibrium position. Q13. A light rigid rod of length L = 0.8 m is able to swing in a vertical plane about a pivot P at a distance L/4 from one end (see Figure). At the lower end is a concentrated mass M = 5 g. The top of the rod is joined by two springs of stiffness = 200 N m 1 to fixed supports. L/4 P By considering the combined torque of all elastic and gravitational forces (or otherwise), calculate the period of small oscillations. If the rod is turned an angle 0.2 radians from the vertical and then released, calculate the maximum speed of the mass M. 3L/4 M Mechanics Examples for Topic E (Oscillations) - 4 David Apsley

5 Q14. A wheel of mass 40 g and radius 0.5 m is able to rotate about a fixed axis through its centre and has radius of gyration 0.4 m about that axis. It is attached to a vertical wall by a spring of stiffness = 1200 N m 1 attached to a point L = 0.25 m from the axis. The system is initially in the position shown, with the spring perpendicular to the radius through its point of attachment. The wheel is then rotated by a small angle θ radians. L (d) (e) What is the moment of inertia of the wheel? Write an expression for the extension of the spring when the wheel has turned through angle θ using the small-angle approximation. Using the angular-momentum principle write down the equation of motion in terms of the angle θ. Determine the circular frequency and period of small oscillations. Show how the assumption of constant total energy also leads to the same equation of motion in terms of the angle θ. Q15. (Exam 2010) When full, a hanging baset can be idealised as a solid hemisphere with mass 6 g and radius 0.15 m. The baset is suspended from a fixed point P by several light chains of length 0.25 m which connect to the rim of the baset (see figure). P 0.25 m 0.15 m Find the moment of inertia of the baset about a horizontal axis through: (i) the centre of mass; (ii) the point P. If the baset is displaced slightly by an angle θ to the vertical through P, show that the subsequent motion is approximately simple harmonic and find the period of small oscillations. If the baset is pulled aside so that angle θ = 0.5 radians, find the maximum angular velocity in the subsequent motion. 6 g For a uniform solid hemisphere of radius R, the centre of mass G lies a distance from the middle of the base, O; the radius of gyration about O is. G O R Mechanics Examples for Topic E (Oscillations) - 5 David Apsley

6 Q16. (Exam 2011) A pendulum consists of a symmetric plane lamina with the dimensions shown in the figure to the right, which is allowed to pivot smoothly about a horizontal axis through the point O. Find: (i) the distance of the centre of mass (G) from O; G (ii) the radius of gyration about O. The pendulum is rotated a small angle from the vertical and released. By writing down its angular-momentum equation, show that its subsequent motion is approximately simple harmonic and determine the period of small oscillations. If the pendulum is first rotated by 90 (so that OG is 0.6 m horizontal) and then released from rest, use energy methods to find its maximum angular velocity in the subsequent motion. The second moment of area of a rectangular lamina, dimensions L W about an axis through its centre and perpendicular to its plane is, where A = LW is its area. 0.2 m O 0.2 m 0.8 m 0.4 m Q17. (Exam 2012) In the system shown below a bar of mass 5 g can pivot smoothly about a horizontal axis through point O, a distance 0.4 m to the left of its centre of mass. The moment of inertia of the bar about this axis is 14 g m 2. Masses 8 g and 5 g are suspended from the ends of the bar by smoothly-pinned light rods connected to the main bar at distances 0.4 m and 1.2 m respectively from the axis whilst the masses are connected to the ground by light springs with stiffnesses 900 N m 1 and 600 N m 1 respectively as shown. When the bar is horizontal the springs are unstretched. Write expressions for the downward displacement of the left-hand mass and the upward displacement of the right-hand mass when the bar is rotated a small angle θ anticlocwise about O. Denoting the tensions in the left and right rods by T 1 and T 2 respectively, write down the equation of linear motion for each of the two masses and the equation of rotational motion of the bar in terms of the angle θ. Show that, with the small-angle approximation, the system can undergo simple harmonic motion and find the period of oscillation. 0.4 m 1.2 m O 5 g 8 g 5 g 900 N/m 600 N/m Mechanics Examples for Topic E (Oscillations) - 6 David Apsley

7 Damped Free Vibrations Q18. (Meriam and Kraige) The cannon fires a 4.5 g cannonball with velocity of 250 m s 1 at 20 to the horizontal. The combined mass of the cannon and its cart are 750 g. If the recoil mechanism consists of a spring of constant = 27 N m 1 and a damper with viscous coefficient c = 9000 N s m 1, determine the maximum recoil deflection x max of the cannon unit. c m 20 o 250 m/s Forced Vibrations c = 1080 N/m 30 g F = 25 cos t Q19. (Meriam and Kraige) The 30 g cart is acted upon by a harmonic force newtons as shown. Determine the ranges(s) of the driving frequency Ω for which the amplitude of the steady-state response is less than 75 mm if c taes the values 0 and 36 N s m 1. Q20. (Exam 2014) A mass 2 g hangs in the loop of a smooth inelastic rope, which is secured to the ceiling by a spring of stiffness = 30 N m 1 as shown. If the mass is pulled down a small distance y, by how much is the spring lengthened? By writing the equation of motion for the mass, show that it can undergo SHM and find the equilibrium extension of the spring and the period of oscillation. 2 g = 30 N/m F = 5 sin 7t (part ) A vertical force newtons is applied to the mass, where t is time in seconds. Find the amplitude of steady oscillation at the forcing frequency. Mechanics Examples for Topic E (Oscillations) - 7 David Apsley

8 Electrical analogue for interest only! Q21. L V(t) R C In a simple electric circuit consisting of inductance L, resistance R and capacitance C, the sum of the potential drops across each component equals the supply voltage V(t); i.e. where I is current and Q is the charge stored on the capacitor. Given that (current = rate of flow of charge), this can be written (*) By comparing this with the forced, damped, mass-spring mechanical system: state what plays the role of: (i) displacement and velocity ; (ii) mass (inertia) (iii) stiffness (iv) damping (friction) (v) force in an electric circuit. Assuming these, (vi) write down an expression for energy in the electric circuit. Write down the simplified form of (*) for no resistance (R = 0) and supply voltage cut off (V = 0). Hence, find the natural frequency of this L C circuit. An alternating supply voltage may be represented by (Since the system is linear we can solve in complex exponentials and then tae the real part as the physical solution). Find a particular integral of (*) in the form (i.e. substitute this and find Q 0 ). Comment on how the amplitude of the response changes as the frequency ω varies. Mechanics Examples for Topic E (Oscillations) - 8 David Apsley