The 2 kg block in Figure 7-25 slides down a frictionless curved ramp, starting from rest at a height of h = 3 m. The block then slides d = 12 m on a rough horizontal surface before coming to rest.
Figure 7-25
(a) What is the speed of the block at the bottom of the ramp?
(b) What is the energy dissipated by friction?
(c) What is the coefficient of friction between the block and the horizontal surface?
a)
(1/2) m v^2 = m g h
so
v = sqrt(2 g h)
b)
m g h
c)
mu m g * 12 = m g h
so
mu = h/12 = 3/12 = .25
(a) Well, the block must be pretty slick if it's sliding down a frictionless ramp! So, let's analyze the situation. We can start by using the conservation of energy. The block starts at a height of 3 m, so its potential energy is given by mgh, where m is the mass (2 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (3 m). The potential energy is then (2 kg)(9.8 m/s^2)(3 m) = 58.8 J.
Since the ramp is frictionless, all of the potential energy is converted into kinetic energy at the bottom. So, the kinetic energy at the bottom of the ramp is also 58.8 J.
Now, we can use the equation for kinetic energy to find the speed. The kinetic energy is given by (1/2)mv^2, where v is the speed at the bottom. Rearranging the equation, we have v = sqrt((2KE)/m). Plugging in the values, we get v = sqrt((2 * 58.8 J) / 2 kg) = sqrt(58.8 J) = 7.67 m/s (approximately).
So, the speed of the block at the bottom of the ramp is approximately 7.67 m/s.
(b) Ah, the energy dissipated by friction, the party pooper of physics! Since the block comes to rest after sliding 12 m on the rough horizontal surface, all of its kinetic energy is dissipated by friction. The initial kinetic energy is 58.8 J (as we found in part (a)).
Therefore, the energy dissipated by friction is 58.8 J.
(c) To find the coefficient of friction between the block and the horizontal surface, we need to use the work-energy principle. The work done by friction is equal to the force of friction times the displacement. In this case, the work done by friction is also equal to the change in kinetic energy (since the block comes to rest).
The work done by friction is given by W = fd, where f is the force of friction and d is the displacement (12 m).
The change in kinetic energy is given by KEf - KEi, where KEf is the final kinetic energy (0 J since the block comes to rest) and KEi is the initial kinetic energy (58.8 J).
Setting up these equations, we have f * 12 m = 0 J - 58.8 J. Since the block comes to rest, the frictional force is equal to the initial kinetic energy, so we have f * 12 m = -58.8 J.
Since we're interested in the coefficient of friction, we can write f as μN, where μ is the coefficient of friction and N is the normal force (equal to the gravitational force since the block is not accelerating vertically). We have μN * 12 m = -58.8 J.
Now, we know the mass of the block (2 kg) and the acceleration due to gravity (9.8 m/s^2). Therefore, N = mg = (2 kg)(9.8 m/s^2) = 19.6 N.
Substituting this into the equation, we have μ * 19.6 N * 12 m = -58.8 J. Solving for μ, we get μ = -58.8 J / (19.6 N * 12 m) ≈ -0.25.
Therefore, the coefficient of friction between the block and the horizontal surface is approximately -0.25. But remember, this negative value means that the force of friction is acting in the opposite direction of motion, slowing down the block. That's a real party pooper!
To determine the speed of the block at the bottom of the ramp (part a), we can use the principle of conservation of energy. The potential energy at the top of the ramp is equal to the kinetic energy at the bottom of the ramp.
Step 1: Calculate the potential energy at the top of the ramp using the formula:
PE = m * g * h
Where m is the mass of the block (2 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (3 m).
PE = 2 kg * 9.8 m/s^2 * 3 m
PE = 58.8 J
Step 2: Calculate the kinetic energy at the bottom of the ramp using the formula:
KE = 1/2 * m * v^2
Where KE is the kinetic energy, m is the mass of the block (2 kg), and v is the velocity.
KE = 58.8 J
Step 3: Solve for v using the equation KE = 1/2 * m * v^2:
58.8 J = 1/2 * 2 kg * v^2
58.8 J = v^2
v^2 = 29.4
v = √29.4
v ≈ 5.42 m/s
Therefore, the speed of the block at the bottom of the ramp is approximately 5.42 m/s.
To determine the energy dissipated by friction (part b), we need to calculate the work done by friction on the block as it slides on the rough horizontal surface.
Step 4: Calculate the work done by friction using the formula:
Work = force * distance
Since the only force acting on the block on the horizontal surface is friction, the work done by friction is equal to the energy dissipated.
Step 5: Find the force of friction using the formula:
force of friction = coefficient of friction * normal force
The normal force is equal to the weight of the block, which is m * g.
So, the force of friction = coefficient of friction * m * g.
Step 6: Determine the distance traveled on the rough surface (d = 12 m).
Step 7: Solve for the energy dissipated by friction using the formula:
Energy dissipated = force of friction * distance
Energy dissipated = coefficient of friction * m * g * d
To find the coefficient of friction (part c), we can use the equation from step 5 above and rearrange it to solve for the coefficient of friction:
coefficient of friction = force of friction / (m * g)
I hope this helps! Let me know if you have any further questions.
To solve these questions, we can use the principles of conservation of energy and the work-energy theorem. Here's how we can approach each part:
(a) To find the speed of the block at the bottom of the ramp, we can use the principle of conservation of mechanical energy. At the top of the ramp, the block only has potential energy given by mgh, where m is the mass of the block, g is the acceleration due to gravity, and h is the height of the ramp.
The potential energy is then converted to kinetic energy as the block slides down the ramp, so we can equate the two:
mgh = 1/2 mv^2
Where v is the velocity (speed) of the block at the bottom of the ramp. We can solve this equation for v by canceling the mass m on both sides and taking the square root:
v = sqrt(2gh)
Substituting the given values:
v = sqrt(2 * 9.8 m/s^2 * 3 m)
v ≈ sqrt(58.8) m/s
v ≈ 7.67 m/s
Therefore, the speed of the block at the bottom of the ramp is approximately 7.67 m/s.
(b) The energy dissipated by friction can be calculated using the work-energy theorem. The theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done by friction is equal to the change in kinetic energy of the block as it comes to rest on the rough horizontal surface.
The initial kinetic energy at the bottom of the ramp is 1/2 mv^2. Since the block comes to rest on the rough surface, the final kinetic energy is zero. Therefore, the work done by friction is equal to the initial kinetic energy:
Work done by friction = 1/2 mv^2
Substituting the given values:
Work done by friction = 1/2 * 2 kg * (7.67 m/s)^2
Work done by friction ≈ 59.14 J
Therefore, the energy dissipated by friction is approximately 59.14 J.
(c) The coefficient of friction between the block and the horizontal surface can be calculated using the formula:
Coefficient of friction (μ) = (Energy dissipated by friction) / (Normal force)
The normal force is equal to the weight of the block (mg), where m is the mass and g is the acceleration due to gravity.
Substituting the given values:
Coefficient of friction = (59.14 J) / ((2 kg) * (9.8 m/s^2))
Coefficient of friction ≈ 3.03
Therefore, the coefficient of friction between the block and the horizontal surface is approximately 3.03.