As shown in the figure below, a 2.25-kg block is released from 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 15.0 cm that has a coefficient of kinetic friction μk = 0.750. Find h such that the block's speed after crossing the rough patch is 4.00 m/s.
First, let's analyze the block's motion when sliding down the ramp. By the conservation of mechanical energy, we have
1/2 * m * v^2 = m * g * h
where m is the mass of the block (2.25 kg), v is the velocity at the bottom of the ramp, g is the acceleration due to gravity (9.8 m/s²), and h is the height of the ramp.
Now let's analyze the block's motion on the surface after leaving the ramp. The work done by the friction force is
W = F_friction * d = μk * m * g * d
where F_friction is the friction force, d is the width of the rough patch, and μk is the coefficient of kinetic friction. The work-energy principle states that the initial kinetic energy of the block plus the work done by the friction force equals the final kinetic energy:
1/2 * m * v_initial^2 + W = 1/2 * m * v_final^2
where v_initial is the initial velocity of the block after leaving the ramp and v_final is the final velocity after crossing the rough patch (4.00 m/s). We can plug in the expression for the work W from above to get
1/2 * m * v_initial^2 + μk * m * g * d = 1/2 * m * v_final^2.
Now we can substitute the expression for v_initial^2 from the conservation of mechanical energy equation and solve for h:
1/2 * m * (2 * g * h) + μk * m * g * d = 1/2 * m * v_final^2
m * g * h + μk * m * g * d = 1/2 * m * v_final^2
Dividing by m and rearranging gives
h = (1/2 * v_final^2 - μk * g * d) / g
Now we can plug in the given values to find h:
h = (1/2 * (4.00 m/s)^2 - 0.750 * 9.8 m/s² * 0.15 m) / (9.8 m/s²)
h ≈ 0.412 m
So the height of the ramp should be about 0.412 meters for the block's speed after crossing the rough patch to be 4.00 m/s.
To find the height h, we can use the principle of conservation of mechanical energy. The initial potential energy when the block is at height h is converted into the final kinetic energy after crossing the rough patch.
1. Write down the formulas for potential energy and kinetic energy:
Potential energy (PE) = mgh
Kinetic energy (KE) = (1/2)mv^2
where m is the mass of the block, g is the acceleration due to gravity, h is the height, and v is the final speed.
2. Set up the energy equation:
Initial potential energy = Final kinetic energy + Energy lost due to friction
mgh = (1/2)mv^2 + Work done against friction
3. Simplify the equation:
Cancel out the mass "m" and divide both sides by 2:
gh = v^2/2 + Work done against friction
4. Find the work done against friction:
Work done against friction = Fd
where F is the frictional force and d is the distance over which it acts.
The frictional force (F) = 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:
N = mg
Therefore, the frictional force is:
F = μk * mg
The distance over which the frictional force acts is the rough patch width:
d = 0.15 m (15.0 cm = 0.15 m)
Plug in the values:
Work done against friction = Fd = (μk * mg) * d
5. Substituting the values and solving for h:
gh = v^2/2 + (μk * mg) * d
Rearranging the equation for h:
h = (v^2/2g) + (μk * d)
Plug in the values:
h = (4.00^2 / 2*9.8) + (0.750 * 0.15)
h = 1.6327 + 0.1125
h = 1.7452 m
Therefore, the height (h) required for the block's speed to be 4.00 m/s after crossing the rough patch is approximately 1.7452 meters.
To find the height h such that the block's speed after crossing the rough patch is 4.00 m/s, we need to use principles of conservation of mechanical energy and the work-energy theorem.
Here's how you can approach this problem:
1. Start by finding the gravitational potential energy of the block at the top of the ramp. We can use the formula: PE = mgh, where m is the mass of the block (2.25 kg), g is the acceleration due to gravity (9.81 m/s^2), and h is the height of the ramp.
2. Since the ramp is frictionless, the block's initial potential energy at the top will be converted entirely into kinetic energy at the bottom of the ramp. So, at the bottom of the ramp, the block will have a kinetic energy equal to its initial potential energy.
3. Write the equation for the kinetic energy of the block at the bottom of the ramp. We can use the formula: KE = (1/2)mv^2, where m is the mass of the block (2.25 kg), and v is the speed of the block at the bottom of the ramp.
4. Set the initial potential energy equal to the kinetic energy at the bottom of the ramp and solve for v. This will give you the speed of the block at the bottom of the ramp.
5. Now, consider the block crossing the rough patch. Calculate the work done by the kinetic friction force on the block during this part of the motion. The work done by friction is given by the formula: W = μk * N * d, where μk is the coefficient of kinetic friction (0.750), N is the normal force acting on the block, and d is the distance over which the block experiences friction (given as the width of the rough patch - 15.0 cm).
6. Use the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy. The work done by friction will cause a decrease in the block's kinetic energy.
7. Write the equation for the work done by friction and set it equal to the change in kinetic energy. Solve for the change in kinetic energy.
8. Subtract the change in kinetic energy from the initial kinetic energy at the bottom of the ramp to find the new kinetic energy after crossing the rough patch. This will be the kinetic energy of the block when its speed is 4.00 m/s.
9. Once you have the new kinetic energy, rewrite the equation for kinetic energy using the speed 4.00 m/s and solve for v.
10. Now, write the equation for kinetic energy using the mass of the block, v, and solve for h.
By following these steps, you should be able to find the value of h that satisfies the given conditions.