The figure shows a 1.50kg block at rest on a ramp of height, h. When the block is released, it reaches the bottom of the ramp and moves across a surface that is frictionless except for one section of width 10cm that has a coefficient of kinetic friction uk=0.64. Find h such that the block's speed after crossing the rough patch is 3.50m/s (Remember that the force of friction is fk=(uk)(N), where N is the normal).
To find the height, h, we can consider the conservation of energy for the block as it moves down the ramp and crosses the rough patch.
1. First, let's determine the potential energy of the block at the top of the ramp. The potential energy is given by the equation:
PE = m * g * h
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.
2. Next, let's consider the work done by gravity on the block as it moves down the ramp. The work done is equal to the change in potential energy:
Work = ΔPE
The change in potential energy is given by:
ΔPE = PE_final - PE_initial
The final potential energy at the bottom of the ramp is zero since the block has reached the bottom. Therefore, the change in potential energy is equal to the initial potential energy:
ΔPE = PE_initial = m * g * h
3. Now, let's consider the work done by friction on the block as it moves across the rough patch. The work is given by the equation:
Work = force_friction * distance
The frictional force can be calculated as:
force_friction = uk * N
Where uk is the coefficient of kinetic friction (0.64), and N is the normal force exerted by the block on the rough surface.
The normal force can be calculated as:
N = m * g
4. The distance the block moves across the rough patch is given as 10 cm, which we'll convert to meters:
distance = 0.10 m
5. Since the frictional force opposes the motion of the block, it does negative work. Therefore, the work done by friction is:
Work = -force_friction * distance
6. Finally, using the work-energy theorem, we can equate the work done by gravity (ΔPE) and the work done by friction:
m * g * h = -force_friction * distance
Substituting the values we have:
1.50 kg * 9.8 m/s² * h = -0.64 * (1.50 kg * 9.8 m/s²) * 0.10 m
Solving for h:
h = -0.64 * 0.10 / 1.50 m = -0.0427 m
However, we need to take into account that h is a height and must be positive. Therefore, we take the absolute value:
h = 0.0427 m
So, the height, h, such that the block's speed after crossing the rough patch is 3.50 m/s, is approximately 0.0427 m.