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 = 0.630. Find h such that the block's speed after crossing the rough patch is 3.20 m/s.
To find the height h, we can use the principles of energy conservation.
First, let's calculate the potential energy of the block at the top of the ramp, at height h. The potential energy (PE) is given by the formula PE = m * g * h, where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s²). In this case, m = 1.50 kg.
PE = 1.50 kg * 9.8 m/s² * h
Next, let's find the total work done on the block. The work done can be broken down into two parts: the work done by gravity while the block slides down the ramp and the work done by friction on the rough patch.
The work done by gravity is equal to the change in potential energy, which is given by m * g * h:
Work_gravity = m * g * h
The work done by friction is equal to the force of friction multiplied by the distance over which it acts. The force of friction (F_friction) can be calculated using the formula F_friction = µk * N, where µk is the coefficient of kinetic friction and N is the normal force.
The normal force (N) is equal to the weight of the block, which is given by m * g:
N = m * g
Work_friction = F_friction * d, where d is the distance over which the friction acts. In this case, the distance is 10.0 cm, which is equal to 0.10 m.
Work_friction = µk * N * d
Now, let's equate the work done by gravity to the work done by friction.
m * g * h = µk * m * g * d
We can cancel out the mass and gravity terms on both sides of the equation:
h = µk * d
Finally, substitute the given values into the equation to find the value of h:
h = 0.63 * 0.10 m
= 0.063 m
Therefore, the height h is approximately 0.063 meters or 6.3 cm.