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A particle moving in the x-y plane is subject to a central force F = f(r)Ã, where r = √x² + y² and F = cos x + sin 0ỹ. = (i) Write the velocity in terms of the polar coordinates r and 0, and thus find an expression for the angular momentum of the particle about the origin, also in terms of the polar coordinates. (ii) Show explicitly that the angular momentum is a constant of the motion. (iii) Derive an expression for the area dA swept out by the particle's position vector r during a small time interval dt (during which its angular displace- ment is de). Hence show that dA/dt is constant (Kepler's Third Law).

A particle moving in the x-y plane is subject to a central force F = f(r)Ã, where r = √x² + y² and F = cos x + sin 0ỹ. = (i) Write the velocity in terms of the polar coordinates r and 0, and thus find an expression for the angular momentum of the particle about the origin, also in terms of the polar coordinates. (ii) Show explicitly that the angular momentum is a constant of the motion. (iii) Derive an expression for the area dA swept out by the particle's position vector r during a small time interval dt (during which its angular displace- ment is de). Hence show that dA/dt is constant (Kepler's Third Law).

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A particle moving in the x-y plane is subject to a central force F = f(r)Ã, where r = √x² + y² and F = cos x + sin 0ỹ. = (i) Write the velocity in terms of the polar coordinates r and 0, and thus find an expression for the angular momentum of the particle about the origin, also in terms of the polar coordinates. (ii) Show explicitly that the angular momentum is a constant of the motion. (iii) Derive an expression for the area dA swept out by the particle's position vector r during a small time interval dt (during which its angular displace- ment is de). Hence show that dA/dt is constant (Kepler's Third Law).
Transcribed Image Text:A particle moving in the x-y plane is subject to a central force F = f(r)Ã, where r = √x² + y² and F = cos x + sin 0ỹ. = (i) Write the velocity in terms of the polar coordinates r and 0, and thus find an expression for the angular momentum of the particle about the origin, also in terms of the polar coordinates. (ii) Show explicitly that the angular momentum is a constant of the motion. (iii) Derive an expression for the area dA swept out by the particle's position vector r during a small time interval dt (during which its angular displace- ment is de). Hence show that dA/dt is constant (Kepler's Third Law).
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