a) The speed of the block at a height of 0.25 m is 2.38 m/s
b) The compression of the spring is 0.25 m
c) The final height of the block is 0.54 m
Explanation:
a)
We can solve the problem by using the law of conservation of energy. In fact, the total mechanical energy (sum of kinetic+gravitational potential energy) must be conserved in absence of friction. So we can write:
[tex]U_i +K_i = U_f + K_f[/tex]
where
[tex]U_i[/tex] is the initial potential energy, at the top
[tex]K_i[/tex] is the initial kinetic energy, at the top
[tex]U_f[/tex] is the final potential energy, at halfway
[tex]K_f[/tex] is the final kinetic energy, at halfway
The equation can be rewritten as
[tex]mgh_i + \frac{1}{2}mu^2 = mgh_f + \frac{1}{2}mv^2[/tex]
where:
m = 2.7 kg is the mass of the block
[tex]g=9.8 m/s^2[/tex] is the acceleration of gravity
[tex]h_i = 0.54[/tex] is the initial height
u = 0 is the initial speed
[tex]h_f = 0.25 m[/tex] is the final height of the block
v is the final speed when the block is at a height of 0.25 m
Solving for v,
[tex]v=\sqrt{u^2+2g(h_i-h_f)}=\sqrt{0+2(9.8)(0.54-0.25)}=2.38 m/s[/tex]
b)
The total mechanical energy of the block can be calculated from the initial conditions, and it is
[tex]E=K_i + U_i = 0 + mgh_i = (2.7)(9.8)(0.54)=14.3 J[/tex]
At the bottom of the ramp, the gravitational potential energy has become zero (because the final heigth is zero), and all the energy has been converted into kinetic energy. However, then the block compresses the spring, and the maximum compression of the spring occurs when the block stops: at that moment, all the energy of the block has been converted into elastic potential energy of the spring. So we can write
[tex]E=E_e = \frac{1}{2}kx^2[/tex]
where
k = 453 N/m is the spring constant
x is the compression of the spring
And solving for x, we find
[tex]x=\sqrt{\frac{2E}{k}}=\sqrt{\frac{2(14.3)}{453}}=0.25 m[/tex]
c)
If there is no friction acting on the block, we can apply again the law of conservation of energy. This time, the initial energy is the elastic potential energy stored in the spring:
[tex]E=E_e = 14.3 J[/tex]
while the final energy is the energy at the point of maximum height, where all the energy has been converted into gravitational potetial energy:
[tex]E=U_f = mg h_f[/tex]
where [tex]h_f[/tex] is the maximum height reached. Solving for this quantity, we find
[tex]h_f = \frac{E}{mg}=\frac{14.3}{(2.7)(9.8)}=0.54 m[/tex]
which is the initial height: this is correct, because the total mechanical energy is conserved, so the block must return to its initial position.
Learn more about kinetic and potential energy:
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