An 8.67-kg block slides with an initial speed of 1.50 m/s down a ramp inclined at an angle of 27.1° with the horizontal. The coefficient of kinetic friction between the block and the ramp is 0.65. Use energy conservation to find the distance the block slides before coming to rest.
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Don't worry! I'll guide you step-by-step through this problem.
We will use the principle of conservation of mechanical energy to solve this problem. The mechanical energy of the block is conserved, which means that the initial mechanical energy should be equal to the final mechanical energy when the block comes to rest.
Let's define the initial mechanical energy (Ei) and the final mechanical energy (Ef) of the block:
Ei = Kinetic energy + Potential energy
Ef = 0 (since the block comes to rest)
Now, let's calculate the initial kinetic energy (Ki) of the block:
Ki = (1/2) * mass * velocity^2
= (1/2) * 8.67 kg * (1.50 m/s)^2
Next, let's calculate the initial potential energy (Pi) of the block:
Pi = mass * gravity * height
= 8.67 kg * 9.8 m/s^2 * (height)
The height can be determined by using the angle of the ramp:
height = (distance) * sin(angle)
Now, let's calculate the distance (d) the block slides before coming to rest:
Since the block comes to rest, the final kinetic energy (Kf) is 0. Now, let's calculate the final potential energy (Pf):
Pf = mass * gravity * height
Since the energy is conserved, we can equate the initial mechanical energy (Ei) to the final mechanical energy (Ef):
Ei = Ef
Ki + Pi = 0
Substituting the formulas for Ki and Pi, we have:
(1/2) * 8.67 kg * (1.50 m/s)^2 + 8.67 kg * 9.8 m/s^2 * (distance * sin(angle)) = 0
Simplifying the equation, we can solve for the distance (d):
(d * sin(angle)) = - (1/2) * (1.50 m/s)^2 / 9.8 m/s^2
distance = (- (1/2) * (1.50 m/s)^2 / 9.8 m/s^2) / sin(angle)
Now, substitute the given values:
distance = (- (1/2) * (1.50 m/s)^2 / 9.8 m/s^2) / sin(27.1°)
Simply evaluate the equation to find the distance the block slides before coming to rest.
Don't worry! I'm here to help you understand how to tackle this problem step by step.
To find the distance the block slides before coming to rest, we can use the concept of energy conservation. Energy conservation states that the total mechanical energy (the sum of kinetic and potential energy) of a system remains constant if no external forces are acting.
In this problem, the only external force acting on the block is the force of kinetic friction opposing its motion. Let's break down the problem into smaller steps:
Step 1: Calculate the gravitational force acting on the block.
The gravitational force can be calculated using the formula: F_gravity = m * g, where m is the mass of the block (8.67 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F_gravity = 8.67 kg * 9.8 m/s^2
Step 2: Calculate the component of the gravitational force acting parallel to the inclined plane.
Since the ramp is inclined at an angle of 27.1°, only a component of the gravitational force is acting parallel to the ramp. We can calculate this component using the formula: F_parallel = F_gravity * sin(theta), where theta is the angle of inclination (27.1°).
F_parallel = F_gravity * sin(theta)
Step 3: Calculate the force of kinetic friction.
The force of kinetic friction can be calculated using the formula: F_kinetic_friction = coefficient of kinetic friction * F_normal, where F_normal is the normal force acting on the block. On an inclined plane, F_normal is equal to mg * cos(theta).
F_normal = m * g * cos(theta)
F_kinetic_friction = coefficient of kinetic friction * F_normal
Step 4: Calculate the work done by the force of kinetic friction.
The work done by the force of kinetic friction can be calculated using the formula: work = force * distance. In this case, the work done by the force of kinetic friction is equal to the negative change in the block's mechanical energy.
work = -ΔKE (change in kinetic energy)
Step 5: Set up the equation for energy conservation.
In this case, the initial mechanical energy is the kinetic energy of the block, and the final mechanical energy is zero (since the block comes to rest).
Initial mechanical energy = Final mechanical energy
(1/2) * m * v_initial^2 + work = 0
Step 6: Solve for the distance.
Rearrange the equation from step 5 to solve for the distance (d):
d = -(1/2) * v_initial^2 / (coefficient of kinetic friction * g * cos(theta) * sin(theta))
Now you can plug in the given values for the mass of the block (m), the initial speed (v_initial), the coefficient of kinetic friction, and the angle of inclination (theta) to calculate the distance (d) the block slides before coming to rest.