Question

A spring (k=15.19kN/m)is is compresses 25cm and held in place on a 36.87° incline. A block (M=10kg) is placed on the spring. When the spring is released the block slides up and off the end of the ramp. The block travels 1.12m along the ramp where the co efficient of kinetic friction is 0.300. Determine the maximum vertical displacement of the block after it becomes airborne relative to it's initial position on the spring

Answer 1

The maximum vertical displacement is 2.07 meters.

Step-by-step explanation:

We can solve this problem using energy. Since there is a frictional force acting on the block, we need to consider the work done by this force. So, the initial potential energy stored in the spring is transferred to the block and it starts to move upwards. Let's name the point at which the block leaves the ramp "1" and the highest point of its trajectory in the air "2". Then, we can say that:

`E_0=E_1\\\\U_e_0=K_1+U_g_1+W_f_1`

Where
`U_e_0` is the elastic potential energy stored in the spring,
`K_1` is the kinetic energy of the block at point 1,
`U_g_1` is the gravitational potential energy of the block at point 1, and
`W_f_1` is the work done by friction at point 1.

Now, rearranging the equation we obtain:

`(1)/(2)kx^(2)=(1)/(2)mv_1^(2)+mgh_1+\mu Ns_1`

Where
`k` is the spring constant,
`x` is the compression of the spring,
`m` is the mass of the block,
`v_1` is the speed at point 1,
`g` is the acceleration due to gravity,
`h_1` is the vertical height of the block at point 1,
`\mu` is the coefficient of kinetic friction,
`N` is the magnitude of the normal force and
`s_1` is the displacement of the block along the ramp to point 1.

Since the force is in an inclined plane, the normal force is equal to:

`N=mg\cos\theta`

Where
`\theta` is the angle of the ramp.

We can find the height
`h_1` using trigonometry:

`h_1=s_1\sin\theta`

Then, our equation becomes:

`(1)/(2)kx^(2)=(1)/(2)mv_1^(2)+mgs_1\sin\theta+\mu mgs_1\cos\theta\\\\\implies v_1=\sqrt{(2((1)/(2)kx^(2)-mgs_1\sin\theta-\mu mgs_1\cos\theta))/(m)}=\sqrt{(kx^(2))/(m)-2gs_1(\sin\theta+\mu \cos\theta)}`

Plugging in the known values, we get:

`v_1=\sqrt{((15190N/m)(0.25m)^(2))/(10kg)-2(9.8m/s^(2))(1.12m)(\sin36.87\°+(0.300) \cos36.87\°)}\\\\v_1=8.75m/s`

Now, we can obtain the height from point 1 to point 2 using the kinematics equations. We care about the vertical axis, so first we calculate the vertical component of the velocity at point 1:

`v_1_y=v_1\sin\theta=(8.75m/s)\sin36.87\°=5.25m/s`

Now, we have:

`y=(v_1_y^(2))/(2g)\\\\y=((5.25m/s)^(2))/(2(9.8m/s^(2)))\\\\y=1.40m`

Finally, the maximum vertical displacement
`h_2` is equal to the height
`h_1` plus the vertical displacement
`y`:

`h_2=h_1+y=s_1\sin\theta +y\\\\h_2=(1.12m)\sin36.87\°+1.40m\\\\h_2=2.07m`

It means that the maximum vertical displacement of the block after it becomes airborne is 2.07 meters.