Question

A 25 kg child bounces on a pogo stick.

The pogo stick has a spring with spring constant 2.0x10^4 N/m. When the child makes a nice big bounce, he finds that at the bottom of the bounce he is accelerating upwards at 9.8 m/s^2. How much is the spring compressed?

Answer 1

Concepts and reason

The required concepts to solve this question are force and hook’s law.

From the definition of the force, it is the product of mass and acceleration. If the acceleration of the object is decreased then the force of the system is also reduced it is due to fact that force is directly proportional to the acceleration of the object. Firstly, write the hook’s law equation and net force acting on the child. Calculate the compressed length of the spring.

Fundamentals

From Newton’s second law the expression of the force is equal to,

$F = ma$

Here,$F$is the force,$m$is the mass of the object and $a$is the acceleration of the object.

The expression for hook law is given as;

$F = kx$

Here, $F$is the applied force and $x$is the displacement by the spring and $k$is spring constant.

The elevator is accelerating upwards.

The net force acting on the child is,

$F = m\left( {a + g} \right)$ …… (1)

According to hooks law, the displacement of the spring with force acting on it is given by,$F = kx$ …… (2)

Substitute $kx$ for $F$in equation (1).

$\begin{array}{c}\\kx = m\left( {a + g} \right)\\\\x = \frac{{m\left( {a + g} \right)}}{k}\\\end{array}$ …… (3)

Use equation (3), the compressed length of the spring is written as follows,

$x = \frac{{m\left( {a + g} \right)}}{k}$

Substitute $25{\rm{ kg}}$ for $m$, $2 \times {10^4}{\rm{ N}} \cdot {{\rm{m}}^{ - 1}}$ for $k$and $9.8{\rm{ m}} \cdot {{\rm{s}}^{ - 2}}$for $a,g$in the above equation.

$\begin{array}{c}\\x = \frac{{\left( {25{\rm{ kg}}} \right)\left( {9.8{\rm{ m}} \cdot {{\rm{s}}^{ - 2}} + 9.8{\rm{ m}} \cdot {{\rm{s}}^{ - 2}}} \right)}}{{\left( {2 \times {{10}^4}{\rm{ N}} \cdot {{\rm{m}}^{ - 1}}} \right)}}\\\\ = 0.024{\rm{ m}}\left( {\frac{{{{10}^2}{\rm{ cm}}}}{{1{\rm{ m}}}}} \right)\\\\ = 2.4{\rm{ cm}}\\\end{array}$

Ans:

The compressed length of the spring is $2.4{\rm{ cm}}$.

Answer 2

The compressed length of the spring is $2.4{\rm{ cm}}$.