A 2-kg block A is pushed up against a spring compressing it a distance x = 0.1 m. The block is then released from rest and slides down the 20° incline until it strikes a 1-kg sphere B which is suspended from a 1 m inextensible rope. The spring constant k = 800 N/m, the coefficient of friction

Question

A 2-kg block A is pushed up against a spring compressing it a distance x=0.1 m. The block is then released from rest and slides down the 20° incline until it strikes a 1-kg sphere B which is suspended from a 1 m inextensible rope. The spring constant k=800 N/m, the coefficient of friction between A and the ground is 0.2, the distance A slides from the unstretched length of the spring d=1.5 m and the coefficient of restitution between A and B is 0.8. When [latex]\alpha=40°[/latex] determine (a) the speed of B (b) the tension in the rope.

Accepted Answer

Data: [latex]\quad m_{A}=2 \mathrm{~kg}, \quad m_{B}=1 \mathrm{~kg}, \quad k=800 \mathrm{~N} / \mathrm{m}, x=0.1 \mathrm{~m}, d=1.5 \mathrm{~m}[/latex]

[latex]\mu_{k}=0.2, \quad e=0.8, \quad \theta=20^{\circ}, \quad \alpha=40^{\circ}, \quad l=1.0 \mathrm{~m} [/latex]

Block slides down the incline:

[latex]\begin{aligned}+\nearrow \Sigma F_{y} & =0 \N-m_{A} g \cos \theta & =0 \N & =m_{A} g \cos \theta \& =(2)(9.81) \cos 20^{\circ} \& =18.4368 \mathrm{~N} \F_{f} & =\mu_{k} N=(0.2)(18.4368) \& =3.6874 \mathrm{~N}\end{aligned} [/latex]

Use work and energy. Datum for [latex]V_{g} [/latex] is the impact point near B.

[latex]\begin{gathered}T_{1}=0, \quad\left(V_{1}\right)_{e}=\frac{1}{2} k x_{1}^{2}=\frac{1}{2}(800)(0.1)^{2}=4.00 \mathrm{~J} \\left(V_{1}\right)_{g}=m_{A} g h_{1}=m_{A} g(x+d) \sin \theta=(2)(9.81)(1.6) \sin 20^{\circ}=10.7367 \mathrm{~J} \U_{1 \rightarrow 2}=-F_{f}(x+d)=-(3.6874)(1.6)=-5.8998 \mathrm{~J} \T_{2}=\frac{1}{2} m_{A} \nu_{A}^{2}=\frac{1}{2}(1)\left(\nu_{A}^{2}\right)=1.000 \nu_{A}^{2} \quad V_{2}=0 \T_{1}+V_{1}+U_{1 \rightarrow 2}=T_{2}+V_{2}: \quad 0+4.00+10.7367-5.8998=1.000 \nu_{A}^{2}+0 \\nu_{A}^{2}=8.8369 \mathrm{~m} / \mathrm{s}^{2} \\mathbf{v}_{A}=2.9727 \mathrm{~m} / \mathrm{s} \quad ⦪ 20^{\circ}\end{gathered} [/latex]

Impact: Conservation of momentum.

Both A and B, horizontal components [latex]\overset{+}{\longrightarrow } [/latex] :

[latex]\begin{gathered}m_{A} \nu_{A} \cos \theta+0=m_{A} \nu_{A}^{\prime} \cos \theta+m_{B} \nu_{B} \(2)(2.9727) \cos 20^{\circ}=2 \nu_{A}^{\prime} \cos 20^{\circ}+(1.00) \nu_{B} \quad \text{(1)}\end{gathered} [/latex]

Relative velocities:

[latex]\begin{aligned}\left(\nu_{B}^{\prime}\right)_{n}-\left(\nu_{A}^{\prime}\right)_{n} & =e\left[\left(\nu_{B}\right)_{n}-\left(\nu_{A}\right)_{n}\right] \\nu_{B}^{\prime} \cos \theta-\nu_{A}^{\prime} & =e\left[\nu_{A}-0\right] \\nu_{B}^{\prime} \cos 20^{\circ}-\nu_{A}^{\prime} & =(0.8)(2.9727) \quad \text{(2)}\end{aligned} [/latex]

Solving Eqs. (1) and (2) simultaneously,

[latex]\begin{aligned}& \nu_{A}^{\prime}=1.0382 \mathrm{~m} / \mathrm{s} \& \nu_{B}^{\prime}=3.6356 \mathrm{~m} / \mathrm{s}\end{aligned} [/latex]

Sphere B rises: Use conservation of energy.

[latex]\begin{aligned}T_{1} & =\frac{1}{2} m_{B}\left(\nu_{B}^{\prime}\right)^{2} \quad V_{1}=0 \T_{2} & =\frac{1}{2} m_{B} \nu_{2}^{2} \quad V_{2}=m_{B} g h_{2}=m_{B} g l(1-\cos \alpha) \T_{1}+V_{1} & =T_{2}+V_{2}: \quad \frac{1}{2} m_{B}\left(\nu_{B}^{\prime}\right)^{2}+0=\frac{1}{2} m_{B} \nu_{2}^{2}+m_{B} g(1-\cos ) \\nu_{2}^{2} & =\left(\nu_{B}^{\prime}\right)^{2}-2 g l(1-\cos \alpha) \& =(3.6356)^{2}-(2)(9.81)\left(1-\cos 40^{\circ}\right) \& =8.6274 \mathrm{~m}^{2} / \mathrm{s}^{2}\end{aligned} [/latex]

(a) Speed of B: [latex]\quad \quad \quad \quad \quad \quad \nu_{2}=2.94 \mathrm{~m} / \mathrm{s}\blacktriangleleft[/latex]

(b) Tension in the rope: ρ= 1.00 m

[latex]a_{n}=\frac{\nu_{2}^{2}}{\rho}=\frac{8.6274}{1.00}=8.6274 \mathrm{~m} / \mathrm{s}^{2}[/latex]

[latex]\begin{aligned}+\nwarrow \Sigma F_{n}=m_{B} a_{n} \text { : }\T-m_{B} g \cos \alpha=m_{B} a_{n}\T=m_{B}\left(a_{n}+g \cos \alpha\right)\=(1.0)\left(8.6274+9.81 \cos 40^{\circ}\right)\end{aligned} [/latex]

[latex]T=16.14 \mathrm{~N}\blacktriangleleft[/latex]

Question 13.200: A 2-kg block A is pushed up against a spring compressing it ...

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