Question

You are asked to design spring bumpers for the walls of a parking garage. A freely rolling 1300 kg car moving at 0.62 m/s is to compress the spring no more than 0.072 m before stopping. What should be the force constant of the spring? Assume that the spring has negligible mass.

Answer 1

To calculate the force constant of the spring for the bumpers, use the work-energy principle to set the kinetic energy of the car equal to the work done by the spring. The required spring constant for the parking garage bumpers is approximately 96765 N/m.

Step-by-step explanation:

To determine the force constant of the spring, we can employ the work-energy principle. The work done by the spring (which is also the energy stored in the spring when compressed) should be equal to the kinetic energy of the car when it starts to compress the spring.

The kinetic energy (KE) of the car can be calculated using KE = (1/2)mv², where m is the mass of the car and v is the velocity. The work done by the spring is W = (1/2)kx², where k is the spring constant and x is the compression distance.

Setting these equal gives (1/2)mv² = (1/2)kx². Solving for k yields k = (mv²)/(x²). Substituting the given values, we get k = (1300 kg × (0.62 m/s)²) / (0.072 m)² = 1300 × 0.3844 / 0.005184 = 96764.8 N/m.

Therefore, the required spring constant for the bumpers is approximately 96765 N/m.

Answer 2

`F=3470.2778\ N`

Step-by-step explanation:

Given:

mass of the car,
`m=1300\ kg`

parking speed of the car,
`u=0.62\ m.s^(-1)`

compression of spring bumpers on the walls,
`\delta x=0.072\ m`

Using the equation of motion:

`v^2=u^2+2.a.\delta x`

where:

`v=` final speed of the car
`=0\ m.s^(-1)`

`a=` acceleration of the car while compressing the spring (will be -ve since final velocity tends to zero)

`0^2=0.62^2+2* a* 0.072`

`a=-2.6694\ m.s^(-1)` (negative sign denotes that it is reducing the speed )

Now the force:

`F=m.a`

`F=1300* 2.6694`

`F=3470.2778\ N`