***EXTRA CREDIT*** Figure 8-33 shows a 1.50 kg block at rest on a ramp of height h. When the block is released, it slides without friction to the bottom of the ramp, and then continues across a surface that is frictionless except for a rough patch of width 10.0 cm that has a coefficient of kinetic friction µk = 0.510. Find h such that the block's speed after crossing the rough patch is 3.70 m/s.
0.69
To find the height h such that the block's speed after crossing the rough patch is 3.70 m/s, we can use the principle of conservation of mechanical energy.
The initial potential energy of the block at the top of the ramp is given by:
PE_initial = m * g * h,
where m is the mass of the block (1.50 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the ramp.
The final kinetic energy of the block after crossing the rough patch is given by:
KE_final = 0.5 * m * v_final^2,
where v_final is the final speed of the block (3.70 m/s).
The kinetic energy of the block after sliding down the ramp before crossing the rough patch is given by:
KE_ramp = m * g * h.
Since there is no friction on the ramp, the kinetic energy after sliding down the ramp is equal to the final kinetic energy after crossing the rough patch.
Therefore, we can set up the equation:
m * g * h = 0.5 * m * v_final^2.
Simplifying the equation, we get:
g * h = 0.5 * v_final^2.
Plugging in the given values, we can solve for h:
9.8 * h = 0.5 * (3.70)^2.
Solving for h, we get:
h = (0.5 * (3.70)^2) / 9.8.
Calculating the value, we find:
h ≈ 0.698 m.
So, the height of the ramp h such that the block's speed after crossing the rough patch is 3.70 m/s is approximately 0.698 meters.
To find the height h, we need to apply the principle of conservation of energy.
The first step is to determine the total initial potential energy of the block at the top of the ramp. The block's initial potential energy is given by the equation:
PE_initial = mgh
where m is the mass of the block (1.50 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height of the ramp.
Next, we need to determine the total final energy of the block after crossing the rough patch of the surface. The final energy consists of kinetic energy and potential energy.
The final kinetic energy is given by the equation:
KE_final = 0.5 * m * v²
where v is the final velocity of the block after crossing the rough patch (3.70 m/s).
The final potential energy is given by the equation:
PE_final = m * g * h'
where h' is the height of the block above the rough patch.
Since energy is conserved, the total initial potential energy should be equal to the sum of the final kinetic energy and final potential energy:
PE_initial = KE_final + PE_final
Substituting the equations and rearranging, we have:
mgh = 0.5 * m * v² + m * g * h'
Simplifying and rearranging the equation, we get:
h = (0.5 * v² - g * h') / g
We can substitute the given values into the equation to calculate the height:
h = (0.5 * (3.70 m/s)² - (9.8 m/s²) * h') / (9.8 m/s²)
Simplifying further, we have:
h = (0.5 * 13.69 m²/s² - 9.8 m/s² * h') / 9.8 m/s²
h = (6.845 m²/s² - 9.8 m/s² * h') / 9.8 m/s²
Now, we need to determine the h' (height above the rough patch). Since the block slides without friction, it would lose all its potential energy when sliding on the rough patch. Therefore, the height h' is simply the width of the rough patch, which is 10.0 cm or 0.1 m.
h = (6.845 m²/s² - 9.8 m/s² * 0.1 m) / 9.8 m/s²
Simplifying further, we get:
h = (6.845 m²/s² - 0.98 m²/s²) / 9.8 m/s²
h = 5.865 m²/s² / 9.8 m/s²
h = 0.598 m
Therefore, the height h that needs to be set for the block is 0.598 m (or 59.8 cm).