Question

A 3.00-kg box is sliding along a frictionless horizontal surface with a speed of 1.8 m/s when it encounters a spring. (a) Determine the force constant of the spring, if the box compresses the spring 5.20 cm before coming to rest. N/m (b) Determine the initial speed the box would need in order to compress the spring by 1.60 cm.

Answer 1

a) k= 3594,7 N/m

b) v= 0.55 m/s

Step-by-step explanation:

• a)
• As the surface is horizontal, the only change in energy will be the change in kinetic energy, as the box comes to an stop after compressing the spring.
• As we know that the surface is frictionless also, this change in kinetic energy must be equal to the change in the elastic potential energy of the spring.
• So we can write the following equality:

`\Delta K = \Delta U`

where
`\Delta K = (1)/(2)*m*v^(2)`

and
`\Delta U = (1)/(2) * k* \Delta x^(2)`

• Simplifying and replacing by the values, we get:

`3.00 kg* (1.8 m/s)^(2) = k* (0.052 m) ^(2)`

• Solving for k:

`k = (3.00kg*(1.8m/s)^(2) )/((0.052m)^(2)) = 3594.7 N/m`

• k = 3594.7 N/m

• b)
• For this part, we can just apply the same equality, replacing the value of k by the one we got, and solving for the initial speed v:

`v = \sqrt{(k*\Delta x^(2))/(m) } = \sqrt{(3594.7N/m*(0.016m)^(2) )/(3.00kg)} = 0.55 m/s`

• v = 0.55 m/s