C1. 5 Stated that the Lorentz force acts on bodies, $\vec{F} = q\left[ \vec{v} \times \vec{B} \right]$ | 0.20 |
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C1. 6
The 2nd Newton's law is written for the system of bodies \[ 2m\dot{\vec{v}}_C = q\left[ \left( \vec{v}_1 + \vec{v}_2 \right) \times \vec{B} \right] \]
Remark: The 2nd Newton's law written correctly for a single ball gives a full score |
0.20 |
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C1. 7 Stated that the center of mass doesn't move. (If this conclusion is made without showing math, the previous marks (0.2+0.2) are also given.) | 0.40 |
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C1. 8
The angular form of 2-nd Newton's law is written \[ \dfrac{d\vec{L}}{dt} = q \left[ \vec{R} \times \left[ \vec{v} \times \vec{B} \right] \right] \]
Remark: this equation also may be written as $z$-axis projection |
0.50 |
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C1. 9 Calculated $L_z = \dfrac{qBz^2}{2}$ | 0.80 |
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C1. 10 Stated that $z$ is maximum when $v_z =0$ | 0.20 |
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C1. 11 The geometry is used correctly to find the direction of $\vec{L}$ when $z$ is maximum and written correct expression $L_z=\sqrt{R^2-z^2}mv$ | 0.40 |
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C1. 12 The kinetic energy is constant because the power of the Lorentz force is zero. | 0.20 |
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C1. 13 The velocity of the ball $v$ is constant | 0.20 |
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C1. 14 The biquadratic equation for $z$ is obtained: $\left( \dfrac{qB}{2mv} \right)^2 z^4 + z^2 - R^2=0$ | 0.20 |
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C1. 15 Calculated $z_\text{max} = \dfrac{\sqrt{2}mv}{qB} \sqrt{\sqrt{1+\left(\frac{qBR}{mv} \right)^2 }-1}$ | 0.30 |
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C1. 16 Tangential acceleration is $a_1=0$ because $v=\text{const}$ | 0.20 |
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C1. 17 Radial acceleration is $a_2 = v^2/R$ | 0.30 |
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C1. 18 Component of the acceleration in the 3-rd direction found from the 2-nd Newton's law: \[ ma_3=qvB\dfrac{z_\text{max}}{R}\] | 0.20 |
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C1. 19 Answer for $a$: \[a=\sqrt{\left( \dfrac{v^2}{R} \right)^2 + \left( \dfrac{qvBz_\text{max}}{mR} \right)^2 }\] | 0.20 |
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C1. 20 Minor mistake: missing numerical factor or a non-dimensional typo when rewriting an equation | -0.10 |
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C1. 21 Major mistake: missing symbol or a dimensional typo when rewriting an equation | -0.20 |
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