12.5E: Exercises for Section 12.5

Finding Components of Acceleration & Kepler's Laws

1) Find the tangential and normal components of acceleration for \(\vecs r(t)=t^2\,\hat{\mathbf{i}}+2t \,\hat{\mathbf{j}}\) when \(t=1\).

Answer:

\(a_\vecs{T}=\sqrt{2}, \quad a_\vecs{N}=\sqrt{2}\)

In questions 2 - 8, find the tangential and normal components of acceleration.

2) \(\vecs r(t)=⟨\cos(2t),\sin(2t),1⟩\)

3) \(\vecs r(t)=⟨e^t \cos t,e^t\sin t,e^t⟩\). The graph is shown here:

Answer:

\(a_\vecs{T}=\sqrt{3}e^t, \quad a_\vecs{N}=\sqrt{2}e^t\)

4) \(\vecs r(t)=⟨\frac{2}{3}(1+t)^{3/2}, \frac{2}{3}(1-t)^{3/2},\sqrt{2}t⟩\)

5) \(\vecs r(t)=⟨2t,t^2,\frac{t^3}{3}⟩\)

Answer:

\(a_\vecs{T}=2t, \quad a_\vecs{N}=2\)

6) \(\vecs r(t)=t^2\,\hat{\mathbf{i}}+t^2\,\hat{\mathbf{j}}+t^3\,\hat{\mathbf{k}}\)

7) \(\vecs r(t)=⟨6t,3t^2,2t^3⟩\)

Answer:

\(a_\vecs{T}=\frac{6t +12t^3}{\sqrt{1+t^2+t^4}}, \quad a_\vecs{N}=6\sqrt{\frac{1+4t^2+t^4}{1+t^2+t^4}}\)

8) \(\vecs r(t)=3\cos(2πt)\,\hat{\mathbf{i}}+3\sin(2πt)\,\hat{\mathbf{j}}\)

Answer:

\(a_\vecs{T}=0, \quad a_\vecs{N}=2\sqrt{3}\pi\)

9) Find the tangential and normal components of acceleration for \(\vecs r(t)=a\cos(ωt)\,\hat{\mathbf{i}}+b\sin(ωt)\,\hat{\mathbf{j}}\) at \(t=0\).

Answer:

\(a_\vecs{T}=0, \quad a_\vecs{N}=aω^2\)

10) Suppose that the position function for an object in three dimensions is given by the equation \(\vecs r(t)=t\cos(t)\,\hat{\mathbf{i}}+t\sin(t)\,\hat{\mathbf{j}}+3t\,\hat{\mathbf{k}}\).

a. Show that the particle moves on a circular cone.

b. Find the angle between the velocity and acceleration vectors when \(t=1.5\).

c. Find the tangential and normal components of acceleration when \(t=1.5\).

Answer:

c. \(a_\vecs{T}=0.43\,\text{m/sec}^2, \quad a_\vecs{N}=2.46\,\text{m/sec}^2\)

11) The force on a particle is given by \(\vecs f(t)=(cost)\,\hat{\mathbf{i}}+(sint)\,\hat{\mathbf{j}}\). The particle is located at point \((c,0)\) at \(t=0\). The initial velocity of the particle is given by \(\vecs v(0)=v_0\,\hat{\mathbf{j}}\). Find the path of the particle of mass \(m\). (Recall, \(\vecs F=m\vecs a\).)

Answer:

\(\vecs r(t)=\left(\frac{-1}{m}\cos t+c+\frac{1}{m}\right)\,\hat{\mathbf{i}}+\left(\frac{−\sin t}{m}+\left(v_0+\frac{1}{m}\right)t\right)\,\hat{\mathbf{j}}\)

12) An automobile that weighs 2700 lb makes a turn on a flat road while traveling at 56 ft/sec. If the radius of the turn is 70 ft, what is the required frictional force to keep the car from skidding?

13) Using Kepler’s laws, it can be shown that \(v_0=\sqrt{\frac{2GM}{r_0}}\) is the minimum speed needed when \(\theta=0\) so that an object will escape from the pull of a central force resulting from mass \(M\). Use this result to find the minimum speed when \(\theta=0\) for a space capsule to escape from the gravitational pull of Earth if the probe is at an altitude of 300 km above Earth’s surface.

Answer:

10.94 km/sec

14) Find the time in years it takes the dwarf planet Pluto to make one orbit about the Sun given that a=39.5 A.U.