A block of mass rests on a frictionless incline that makes an angle with the horizontal

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Mathematical Methods for Physics, Ch Vector spaces, bases, linear maps and dual vectors

Problem

A block of mass m=2.0 kgm = 2.0\,\mathrm{kg} rests on a frictionless incline that makes an angle θ=25.0∘\theta = 25.0^\circ with the horizontal. In the lab axes x^\hat{\mathbf x} (horizontal) and y^\hat{\mathbf y} (vertical, upward) the weight is W=(0,−mg)\mathbf{W} = (0, -mg) with g=9.81 m s−2g = 9.81\,\mathrm{m\,s^{-2}}. Define a second orthonormal basis {e^1,e^2}\{\hat{\mathbf e}_1, \hat{\mathbf e}_2\} aligned with the incline, e^1\hat{\mathbf e}_1 pointing down the slope and e^2\hat{\mathbf e}_2 pointing out of the slope along its outward normal: e^1=(cos⁡θ,−sin⁡θ),e^2=(sin⁡θ,cos⁡θ).\hat{\mathbf e}_1 = (\cos\theta, -\sin\theta), \qquad \hat{\mathbf e}_2 = (\sin\theta, \cos\theta). (a) Starting from the fact that {e^1,e^2}\{\hat{\mathbf e}_1,\hat{\mathbf e}_2\} is orthonormal, derive the general formula for the components c1,c2c_1, c_2 of a vector v\mathbf{v} in this basis. (b) Use the formula to find the components of W\mathbf{W} along e^1\hat{\mathbf e}_1 and e^2\hat{\mathbf e}_2, to three significant figures. (c) Check the result by confirming that c12+c22c_1^2+c_2^2 equals ∣W∣2|\mathbf{W}|^2 to three significant figures, carrying the unrounded components into the check.

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