17.2. Energy in simple harmonic motion
A subsection of Physics, 9702, through 17. Oscillations
Listing 9 of 9 questions
A uniform beam is clamped at one end. A metal block of mass m is fixed to the other end of the beam causing it to bend, as shown in . x clamp beam equilibrium position displaced position metal block mass m The block is given a small vertical displacement and then released so that it oscillates with simple harmonic motion. The acceleration a of the block is given by the expression a x m k =- where k is a constant for the beam and x is the vertical displacement of the block from its equilibrium position. Explain how it can be deduced from the expression that the block moves with simple harmonic motion. For the beam, k = 4.0 kg s–2. Show that the angular frequency ω of the oscillations is given by the expression ω . m 2 0 = . The initial amplitude of the oscillation of the block is 3.0 cm. Use the expression in to determine the maximum kinetic energy of the oscillations. maximum kinetic energy = J Over a certain interval of time, the maximum kinetic energy of the oscillations in is reduced by 50%. It may be assumed that there is negligible change in the angular frequency of the oscillations. Determine the amplitude of oscillation. amplitude = m Permanent magnets are now positioned so that the metal block oscillates between the poles, as shown in . beam metal block permanent magnets The block is made to oscillate with the same initial amplitude as in . Use energy conservation to explain why the energy of the oscillations decreases more rapidly than in .
9702_m17_qp_42
THEORY
2017
Paper 4, Variant 2
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A ball of mass 37 g is held between two fixed points A and B by two stretched helical springs, as shown in . ball mass 37 g A B The ball oscillates along the line AB with simple harmonic motion of frequency 3.5 Hz and amplitude 2.8 cm. Show that the total energy of the oscillations is 7.0 mJ. At two points in the oscillation of the ball, its kinetic energy is equal to the potential energy stored in the springs. Calculate the magnitude of the displacement at which this occurs. displacement = cm On the axes of and using your answers in and , sketch a graph to show the variation with displacement x of the total energy of the system (label this line T), the kinetic energy of the ball (label this line K), the potential energy stored in the springs (label this line P). –3 –2 –1 energy / mJ x / cm The arrangement in is now rotated through 90° so that the line AB is vertical and the ball oscillates in a vertical plane. Suggest one form of energy, other than those in , that must be taken into consideration when plotting new graphs to show energy changes with displacement.
9702_s12_qp_42
THEORY
2012
Paper 4, Variant 2
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A heavy metal sphere of mass 0.81 kg is suspended from a string. The sphere is undergoing small oscillations from side to side, as shown in . string heavy sphere, mass 0.81 kg oscillations The oscillations of the sphere may be considered to be simple harmonic with amplitude 0.036 m and period 3.0 s. State what is meant by simple harmonic motion. Calculate: the angular frequency of the oscillations angular frequency = rad s–1 the total energy of the oscillations. total energy = J The suspended sphere is now lowered into water. The sphere is given a sideways displacement of +0.036 m from its equilibrium position and is then released at time t = 0. The water causes the motion of the sphere to be critically damped. On , sketch the variation of the displacement x of the sphere from its equilibrium position with t from t = 0 to t = 6.0 s. 0.02 0.04 – 0.02 – 0.04 x / m t / s
9702_w23_qp_41
THEORY
2023
Paper 4, Variant 1
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A heavy metal sphere of mass 0.81 kg is suspended from a string. The sphere is undergoing small oscillations from side to side, as shown in . string heavy sphere, mass 0.81 kg oscillations The oscillations of the sphere may be considered to be simple harmonic with amplitude 0.036 m and period 3.0 s. State what is meant by simple harmonic motion. Calculate: the angular frequency of the oscillations angular frequency = rad s–1 the total energy of the oscillations. total energy = J The suspended sphere is now lowered into water. The sphere is given a sideways displacement of +0.036 m from its equilibrium position and is then released at time t = 0. The water causes the motion of the sphere to be critically damped. On , sketch the variation of the displacement x of the sphere from its equilibrium position with t from t = 0 to t = 6.0 s. 0.02 0.04 – 0.02 – 0.04 x / m t / s
9702_w23_qp_43
THEORY
2023
Paper 4, Variant 3
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A student investigates the energy changes of a mass oscillating on a vertical spring, as shown in . spring mass The student draws a graph of the variation with displacement x of energy E of the oscillation, as shown in . – 1.5 – 1.0 – 0.5 x / cm 0.5 1.0 1.5 2.0 2.5 1.5 1.0 0.5 E / mJ State whether the energy E represents the total energy, the potential energy or the kinetic energy of the oscillations. The student repeats the investigation but with a smaller amplitude. The maximum value of E is now found to be 1.8 mJ. Use to determine the change in the amplitude. Explain your working. change in amplitude = cm
9702_s14_qp_41
THEORY
2014
Paper 4, Variant 1
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A student investigates the energy changes of a mass oscillating on a vertical spring, as shown in . spring mass The student draws a graph of the variation with displacement x of energy E of the oscillation, as shown in . – 1.5 – 1.0 – 0.5 x / cm 0.5 1.0 1.5 2.0 2.5 1.5 1.0 0.5 E / mJ State whether the energy E represents the total energy, the potential energy or the kinetic energy of the oscillations. The student repeats the investigation but with a smaller amplitude. The maximum value of E is now found to be 1.8 mJ. Use to determine the change in the amplitude. Explain your working. change in amplitude = cm
9702_s14_qp_43
THEORY
2014
Paper 4, Variant 3
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Questions Discovered
9