- Dynamics is the study of the physical laws of motion. - (1.2) Figure 1.2 depicts a small portion of the curve (1.1). - which is of course the area of the trapezium α, β , γ, δ.. - Equation (1.28) is of the form f (v. - The change in the velocity, say ∆v, of the ball over a time. - In terms of the Cartesian coordinates x(t) and z(t) the vector r(t) is x(t). - (a) Show that if one chooses coordinate system for the problem as shown above (x, y), then the equation of the path is. - The subtraction r 2 − r 1 has the meaning of the relative displacement. - Calculate the speed of the car.. - The momenta of the three masses in Figure 1.11 are p 1 = m 1 d. - Determine (a) the acceleration of the system. - which is the change in momentum of the mass m.. - F is equal to the rate of change of momentum of the mass m:. - Describe the resulting motion of the 5m mass.. - The initial and final velocities of the spheres are. - Determine the initial velocities of the particles. - u, where m and M are the masses of the spheres. - (b) Show that the speed of the Moon in its orbit is approximately 1.0 km/s.. - This means that the change in potential energy of the object is − mg · s. - ds = ES cos θ, where S is the total length of the path.. - a · ds is the gain in kinetic energy of the object.. - (e) What is the steepest gradient of the slope?. - (b) Calculate the value of the loop integral. - Hint: You need to show that the force on the object is proportional to the distance from the centre of the Earth. - One part of the object is at position (x, y, z) when its velocity is (u, v, w). - where ω is the angular velocity of the rotating frame.. - Calculate the acceleration of a point on the rim of the millstone:. - where m is the mass of the astronaut. - Looking at the direction ∗ of the force vector. - The constant e is the eccentricity of the conic section.. - where v is the magnitude of the vector v.. - (a) Show that r∆r sin φ is just the area of the parallelogram ABCD.. - Let us just take stock of the major results:. - What is the amplitude of the motion?. - and that the complex amplitude of the combined wave is given by the number A + Be iφ . - (a) Show that the equation satisfied by the motion of the particle is F = ma. - (4.21) (f) In this case, show that the solution is of the form. - Calculate the position of the nodes. - Derive an expression for the amplitude of the oscillations at position x. - of the wave at the detector. - The total disturbance is equal to the sum of the contributing wave displacements:. - We shall start with a wave of the form y = A cos (ωt − kx + φ. - where c = ω/k is the speed of the wave. - Accordingly, all of the interesting motion will be in the vertical direction.. - And so the force is equal to the impedance Z multiplied by the velocity of the waving string. - The impedance of the material is Z L for x <. - The actual shape of the wave for x <. - On the left of the join the force is similarly given by − T ∂(y i +y r )/∂x.. - Q16 Show that the efficiency of the join in terms of power (rather than amplitude) is given by η = 4Z L Z R. - where E 0 is the amplitude of the wave 1 m from the source.. - Q18 The power output of the Sun is about W. - 4πD 2 , where P is the total visible power output (luminosity) of the star. - Assume that all of the current is carried by the copper ions. - The potential difference of the mains supply is about 230 V. - What is the ratio of the potential to kinetic energy?. - terminal of the battery is chosen as the reference point. - The potential of the reference point (measured with respect to itself) is of course zero.. - Label the negative terminal of the battery (top left point) ‘0 V’.. - The impedance of the component is defined as the complex number Z = V 0 /J 0 . - (a) Calculate the amplitude of the current.. - (b) Calculate the amplitude of the electron speed in the connecting wire. - Calculate the amplitude of the voltage.. - Engineers talk of the ‘current leading the voltage’ in a capacitor.. - Engineers talk of the current in an inductor ‘lagging’ the voltage.. - So the voltage drop is in the direction of the current if dI/dt >. - The diagram is basically a drawing of the complex. - ‘leading’ it in the case of the capacitor.. - and the phase of the voltage with respect to the current is given by. - and the overall impedance of the circuit is Series : Z = V. - and the overall impedance of the circuit is given by. - The impedance of the capacitor is − i/ωC. - Accordingly, the impedance of the combi- nation is given by. - From Figure 5.9 we can see that the magnitude of the voltage will be. - After (4) we of course return to (1) and the beginning of the next cycle.. - zero and the temperature of the triple point of water. - Let us suppose that the temperature of the boiler in a steam engine is T A . - This is an alternative definition of the second law. - Q7 Estimate the altitude of the mountaineer in Q6. - Assume that all of the air in the atmosphere is at 0 ◦ C. - We also presented a justification of the Boltzmann law in the form of a workshop.. - F = d dt P = 0, where P is the total momentum of the system.. - The initial mass of the nth stage, M 0 (n. - the structural mass of the nth stage, M s (n. - and the initial mass of the next stage, M 0 (n+1). - (j) It is found that the pressure difference is proportional to the length of the pipe.. - (a) Calculate the relative uncertainty of the following measurements:. - How does this compare with the relative uncertainty of the speed?. - of the body. - (b) Now assume that the density of the cone in Figure 7.4 is a function of y:. - 0, which means that (r × p), the angular momentum of the system must be a constant. - r 4 sin 3 θ dr dθ dϕ, with ρ being the density of the material.. - Workshop: Parallel axes theorem 7.9 155 where R is the radius of the sphere. - dt , where M is the total mass of the sphere.. - Show that the acceleration of the centre of mass of the sphere is given by a = 5. - where d is the magnitude of the displacement d (look at the triangle ABC).. - The eccentricity (e) of the trajectory is related directly to the total energy (E) of the system. - dQ T = 0 Re-statement of the first law for a reversible process. - The extremities of the x-values are obtained by setting y = 0:. - We call this matrix the transpose of the first, R(θ) T. - where T is the time period of the rotation