A 70-kg person rides in a 30-kg cart moving at 12 m/s at the top of a hill that is in the shape of an arc of a circle with a radius of 40 m. (a) What is the apparent weight of the person as the cart passes over the top of the hill? (b) Determine the maximum speed that the cart can travel at the top of the hill without losing contact with the surface. Does your answer depend on the mass of the cart or the mass of the person? Explain.

According to Newton’s second law, the linear force is given by,

F = ma

Here,F is linear force,m is the mass of object, anda is acceleration of object.

The centripetal acceleration is given by,

${\mathbf{a}}_{\mathbf{c}}\mathbf{=}\frac{{\mathbf{V}}^{\mathbf{2}}}{\mathbf{r}}$

Here,v is velocity, andr is radius of curvature.

(a)

Draw the free-body diagram of the give situation as follows.

According to the Newton’s second law, the centripetal force is given by,

$$\begin{gathered} \sum _{}{F}_y=-ma \\ N-mg=-m{a}_c \\ N-mg=-\frac{m{v}^2}{r} \\ N=mg-\frac{m{v}^2}{r}.......\left(1\right) \end{gathered}$$

Substitute 70 kg for m, $9.8m/s^2$ for g, 12 m/s for v, and 40 m forr in equation (1).

$$\begin{gathered} \mathrm{N}=\left(70\mathrm{kg}\right)\left(9.8\mathrm{m}/{\mathrm{s}}^2\right)-\frac{\left(70\mathrm{kg}\right){\left(12\mathrm{m}/\mathrm{s}\right)}^2}{40\mathrm{m}} \\ =686-252 \\ =434\mathrm{N} \end{gathered}$$

Therefore, the apparent weight of the person is 434 N.

(b)

The speed at which the cart will slide off means that N=0. From equation (1),

$$\begin{gathered} 0=\mathrm{mg}-\frac{{\mathrm{mv}}^2}{\mathrm{r}} \\ \mathrm{mg}=\frac{{\mathrm{mv}}^2}{\mathrm{r}} \\ \mathrm{g}=\frac{{\mathrm{v}}^2}{\mathrm{r}} \\ {\mathrm{v}}_{\max }=\sqrt{\mathrm{gr}}..........\left(2\right) \end{gathered}$$

Substitute$9.8\mathrm{m}/{\mathrm{s}}^2$ for g, and 40 m for rin equation (2).

$$\begin{gathered} {\mathrm{v}}_{\max }=\sqrt{\left(9.8\mathrm{m}/{\mathrm{s}}^2\right)\left(40\mathrm{m}\right)} \\ =19.8\mathrm{m}/\mathrm{s} \end{gathered}$$

Therefore, the maximum velocity is 19.8 m/s.

From the equation (2), it can be observed that the maximum velocity does not depend on the mass of the cart or person.