(A) Estimate the magnitude of the force F a person must apply to a wheelchair’s main wheel to roll up over a sidewalk curb (Fig. 12.12a). This main wheel that comes in contact with the curb has a radius r, and the height of the curb is h.

(B) Determine the magnitude and direction of R.

Question

(A) Estimate the magnitude of the force F a person must apply to a wheelchair’s main wheel to roll up over a sidewalk curb (Fig. 12.12a). This main wheel that comes in contact with the curb has a radius r, and the height of the curb is h.

(B) Determine the magnitude and direction of R.

Accepted Answer

Normally, the person’s hands supply the required force to a slightly smaller wheel that is concentric with the main wheel. For simplicity, we assume that the radius of the smaller wheel is the same as the radius of the main wheel.
Let us estimate a combined weight of mg = 1 400 N for the person and the wheelchair and choose a wheel radius of r = 30 cm. We also pick a curb height of h = 10 cm. We assume that the wheelchair and occupant are symmetric, and that each wheel supports a weight of 700 N. We then proceed to analyze only one of the wheels. Figure 12.12b shows the geometry for a single wheel.

When the wheel is just about to be raised from the street, the reaction force exerted by the ground on the wheel at point B goes to zero. Hence, at this time only three forces act on the wheel, as shown in the free-body diagram in Figure 12.12c. However, the force R, which is the force exerted by the curb on the wheel, acts at point A, and so if we choose to have our axis of rotation pass through point A, we do not need to include R in our torque equation. From the triangle OAC shown in Figure 12.12b, we see that the moment arm d of the gravitational force mg acting on the wheel relative to point A is

[latex] d=\sqrt{r^{2}-(r-h)^{2}}=\sqrt{2 r h-h^{2}} [/latex].

The moment arm of F relative to point A is 2r - h (see Figure 12.12c). Therefore, the net torque acting on the wheel about point A is

[latex] m g d-F(2 r-h)=0 [/latex].

[latex] m g \sqrt{2 r h-h^{2}}-F(2 r-h)=0 [/latex].

[latex] F=\frac{m g \sqrt{2 r h-h^{2}}}{2 r-h} [/latex].

[latex] =\frac{(700 N ) \sqrt{2(0.3 m )(0.1 m )-(0.1 m )^{2}}}{2(0.3 m )-0.1 m } [/latex].

[latex] =3 imes 10^{2} N [/latex].

(Notice that we have kept only one digit as significant.) This result indicates that the force that must be applied to each wheel is substantial. You may want to estimate the force required to roll a wheelchair up a typical sidewalk accessibility ramp for comparison.

(B) Determine the magnitude and direction of R.

We use the first condition for equilibrium to determine the direction:

[latex] \sum F_{x}=F-R \cos heta=0 [/latex].

[latex] \sum F_{y}=R \sin heta-m g=0 [/latex].

Dividing the second equation by the first gives

[latex] an heta=\frac{m g}{F}=\frac{700 N }{300 N } [/latex].

[latex] heta=70^{\circ} [/latex].

We can use the right triangle shown in Figure 12.12d to obtain R:

[latex] R=\sqrt{(m g)^{2}+F^{2}}=\sqrt{(700 N )^{2}+(300 N )^{2}}=800 N [/latex].

Question 12.5: (A) Estimate the magnitude of the force F a person must appl...

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