By Kleppner D., Kolenkow R.
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Extra resources for An Introduction to Mechanics
Use a coordinate system with the x axis in the equatorial plane and passing through the prime meridian; let the z axis be on the polar axis, positive toward the north pole, as shown in the sketch. TA Time α g= 8h , T A − T B2 2 R where h is the height of line B above line A. 17 Rolling drum A drum of radius R rolls down a slope without slipping. Its axis has acceleration a parallel to the slope. What is the drum’s angular acceleration α? 18 Elevator and falling marble* At t = 0, an elevator departs from the ground with uniform speed.
5 we demonstrated how to describe velocity and acceleration by vectors. In particular, we showed how to diﬀerentiate the vector r to obtain a new vector v = dr/dt. We will want to diﬀerentiate other vectors with respect to time on occasion, so it is worthwhile generalizing our discussion. 10 MORE ABOUT THE TIME DERIVATIVE OF A VECTOR 23 Consider a vector A(t) that changes with time. The change in A(t) during the interval from t to t + Δt is A(t + Δt ) ΔA ΔA = A(t + Δt) − A(t). 6, we define the time derivative of A by A + ΔA A(t + Δt) − A(t) dA = lim .
1 Vector algebra 1* ˆ and B = (5ˆi + ˆj + 2k) ˆ find: Given two vectors A = (2ˆi − 3ˆj + 7k) (a) A + B; (b) A − B; (c) A · B; (d) A × B. 2 Vector algebra 2* ˆ and B = (6ˆi − 7ˆj + 4k) ˆ find: Given two vectors A = (3ˆi − 2ˆj + 5k) (a) A2 ; (b) B2 ; (c) (A · B)2 . 3 Cosine and sine by vector algebra* ˆ Find the cosine and the sine of the angle between A = (3ˆi + ˆj + k) ˆ ˆ ˆ and B = (−2i + j + k). 4 Direction cosines The direction cosines of a vector are the cosines of the angles it makes with the coordinate axes.