Question

In Fig. 8-21, a small, initially stationary block is released on a frictionless ramp at a height of 3.0 m. Hill heights along the ramp are as shown in the figure. The hills have identical circular tops, and the block does not fly off any hill. (a) Which hill is the first the block cannot cross? (b) What does the block do after failing to cross that hill? Of the hills that the block can cross, on which hill-top is (c) the centripetal acceleration of the block greatest and (d) the normal force on the block least?

Short answer

(a)The block cannot cross the fourth hill.

(b) The block travels back and forth across the path after failing to cross the hill.

(c)On the smallest hilltop, the centripetal acceleration of the block is the greatest

(d) On the first hilltop, the normal force is least.

Step-by-step solution

Given information 

Height of the frictionless ramp$\mathrm{h}=3\hspace{0.33em}\mathrm{cm}$

To understand the concept

The problem deals with the centripetal acceleration. This is a acceleration of a body traversing a circular path. From the height of the ramp and maximum height, the first hill which the block cannot cross can be found. Using the formula of centripetal acceleration, the hilltop on which the centripetal acceleration of the block is the greatest will be calculated.

Formula:

The centripetal acceleration is given by,

$$\begin{gathered} {\mathrm{a}}_{\mathrm{C}}={\mathrm{v}}^2/\mathrm{r} \\ \mathrm{KE}+\mathrm{PE}=\mathrm{constant} \end{gathered}$$

(a) To find the first hill which the block cannot cross

The height of the frictionless ramp is given as$\text{h=3\hspace{0.33em}cm}$.

Maximum height that the block should cross is$\mathrm{H}=3.5\mathrm{cm}$.

As$\mathrm{H}>\mathrm{h}$, block cannot cross the fourth hill.

Therefore, the block cannot cross the fourth hill.

(b) What does the block do after failing to cross the hill?

As$\mathrm{H}>\mathrm{h}$ , the block cannotcross the fourthhill. Therefore, the block fails to cross the fourth hill.

Thus, it travels back and forth across the path over the first three hills.

(c) To find the hilltop on which hilltop is the centripetal acceleration of the block the greatest

It is known that

$\mathrm{KE}+\mathrm{PE}=\mathrm{constant}$

According to the principle of conservation of energy, the gravitational potential energy will be less when kinetic energy is more. Kinetic energy will be more when its speed is more. As the height of the first hill is small, the centripetal acceleration${\mathrm{a}}_{\mathrm{C}}={\mathrm{v}}^2/\mathrm{r}$ will be more.

(d) To find the hilltop on which hilltop is the normal force on the block the least

For the block on the first hilltop

${\mathrm{F}}_{\mathrm{n}}=\mathrm{mg}-\frac{{\mathrm{mv}}^2}{\mathrm{r}}$

Where,

$$\begin{gathered} {\mathrm{F}}_{\mathrm{n}}=\mathrm{normal}\mathrm{force}\mathrm{of}\mathrm{the}\mathrm{block} \\ \mathrm{mg}=\mathrm{gravitational}\mathrm{force}\mathrm{on}\mathrm{the}\mathrm{block} \end{gathered}$$

$\frac{{\mathrm{mv}}^2}{\mathrm{r}}=\mathrm{centripetal}\mathrm{force}$

As the velocity of the block is maximum on the first hilltop, the normal force is minimum on the first hill.