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Magazine Chapter 7: Problem 61
When a 3.0-kg block is pushed against a massless spring of force constant constant \(4.5 \times 10^{3} \mathrm{N} / \mathrm{m},\) the spring is compressed \(8.0 \mathrm{cm} .\) The block is released, and it slides \(2.0 \mathrm{m}\) (from the point at which it is released) across a horizontal surface before friction stops it. What is the coefficient of kinetic friction between the block and the surface?
Short Answer
Expert verified
The coefficient of kinetic friction between the block and the horizontal surface is approximately \(2.44\).
Step by step solution
01
Calculate the spring force
We are given that the force constant of the spring is k = 4.5 x 10^3 N/m. According to Hooke's law, the force exerted by the spring when it is compressed is given by:
F = k * x
where,
F = Spring force
k = Force constant of the spring = 4.5 x 10^3 N/m
x = Compression of the spring = 8.0 cm = 0.08 m
Using these values, we can calculate the spring force:
F = 4.5 x 10^3 N/m * 0.08 m = 360 N
02
Calculate the work done by the spring
Next, we need to find the work done by the spring on the block when it is released. The work done by the spring can be determined using the following formula:
W = (1/2) * k * x^2
where,
W = Work done by the spring
k = Force constant of the spring = 4.5 x 10^3 N/m
x = Compression of the spring = 0.08 m
Now we plug in the values and calculate the work done by the spring:
W = (1/2) * 4.5 x 10^3 N/m * (0.08 m)^2 = 144 J
03
Use the work-energy theorem to find the coefficient of kinetic friction
Now that we have the work done by the spring, we can use the work-energy theorem to find the coefficient of kinetic friction. The work-energy theorem states that the total work done on an object is equal to its change in kinetic energy:
W = 螖KE
In this problem, the work done by the spring is W = 144 J and the change in kinetic energy is given by:
螖KE = (1/2) * m * v^2_final - (1/2) * m * v^2_initial
= (1/2) * m * v^2_final - 0 (since initial velocity is zero)
The block slides 2.0 m across the horizontal surface before friction stops it, so the final velocity is also zero:
螖KE = 0 - (1/2) * m * v^2_initial
The total work done on the block is the sum of the work done by the spring and the work done by friction:
W = W_spring + W_friction
Since the friction opposes the motion, W_friction = -渭 * N * d, where 渭 is the coefficient of kinetic friction, N is the normal force, and d is the distance the block slides.
We know that N = m*g, where m is the mass of the block and g is the acceleration due to gravity, so:
W = W_spring - 渭*m*g*d
Now we plug in the known values and solve for 渭:
144 = 0 - (-渭*3*9.81*2)
144 = 58.86 * 渭
Solving for 渭, we get:
渭 = 144 / 58.86 鈮� 2.44
So the coefficient of kinetic friction between the block and the surface is approximately 2.44.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Work-Energy Theorem
The Work-Energy Theorem is a fundamental concept that connects the work done on an object with its kinetic energy. In simple terms, it states that the total work done on an object is equal to the change in its kinetic energy.
In mathematical form, it is expressed as:
\[ W = \Delta KE \]
where \( W \) is the work done, and \( \Delta KE \) is the change in kinetic energy. This theorem helps to understand how energy is transferred from one form to another, typically from potential to kinetic energy, or vice versa.
For the given exercise, we used this theorem to set up our equation by considering the work done by both the spring and friction forces. Analyzing these energy transformations allows us to solve for the unknown coefficient of kinetic friction.
In mathematical form, it is expressed as:
\[ W = \Delta KE \]
where \( W \) is the work done, and \( \Delta KE \) is the change in kinetic energy. This theorem helps to understand how energy is transferred from one form to another, typically from potential to kinetic energy, or vice versa.
For the given exercise, we used this theorem to set up our equation by considering the work done by both the spring and friction forces. Analyzing these energy transformations allows us to solve for the unknown coefficient of kinetic friction.
Hooke's Law
Hooke's Law describes the behavior of springs in relation to the forces they exert when compressed or stretched. It states that the force exerted by a spring is directly proportional to the amount it is stretched or compressed. Formally, the law is expressed as:
\[ F = -k \cdot x \]
where \( F \) represents the force exerted by the spring, \( k \) is the spring constant, and \( x \) is the displacement from the spring's equilibrium position. The negative sign indicates that the force is a restoring force, acting in the opposite direction of displacement.
In our problem, we applied Hooke's Law to calculate the spring force when the block compressed the spring by 8 cm. We found that the spring exerted a force of 360 N, providing the initial potential energy to the block.
\[ F = -k \cdot x \]
where \( F \) represents the force exerted by the spring, \( k \) is the spring constant, and \( x \) is the displacement from the spring's equilibrium position. The negative sign indicates that the force is a restoring force, acting in the opposite direction of displacement.
In our problem, we applied Hooke's Law to calculate the spring force when the block compressed the spring by 8 cm. We found that the spring exerted a force of 360 N, providing the initial potential energy to the block.
Coefficient of Friction
The coefficient of friction quantifies how much frictional force exists between two surfaces. There are two types: static and kinetic, but here we're focused on the kinetic coefficient, which comes into play once movement has started.
The kinetic friction force \( F_f \) is calculated using:
\[ F_f = \mu_k \cdot N \]
where \( \mu_k \) is the coefficient of kinetic friction and \( N \) is the normal force. This normal force is usually the weight of the object when on a horizontal surface.
In this exercise, determining \( \mu_k \) required understanding the overall energy balance. We used the work-energy relationship and the forces involved to reveal that friction converted the block's kinetic energy into thermal energy, eventually stopping the block. Solving the equations, we found \( \mu_k \approx 2.44 \), the ratio of frictional force resisting the block's motion.
The kinetic friction force \( F_f \) is calculated using:
\[ F_f = \mu_k \cdot N \]
where \( \mu_k \) is the coefficient of kinetic friction and \( N \) is the normal force. This normal force is usually the weight of the object when on a horizontal surface.
In this exercise, determining \( \mu_k \) required understanding the overall energy balance. We used the work-energy relationship and the forces involved to reveal that friction converted the block's kinetic energy into thermal energy, eventually stopping the block. Solving the equations, we found \( \mu_k \approx 2.44 \), the ratio of frictional force resisting the block's motion.
Spring Force
Spring force refers to the force a spring exerts when it is either compressed or stretched. This force is a direct application of Hooke's Law and serves as a potential energy store.
The potential energy stored in a spring when compressed is calculated by:
\[ E_{spring} = \frac{1}{2} \cdot k \cdot x^2 \]
In this problem, this energy is released as kinetic energy when the block is freed from the spring. We calculated the work done by the spring, converting it to kinetic energy as the block pushed back out, allowing us to understand how the spring's stored energy became the driving force for the block's initial motion.
The potential energy stored in a spring when compressed is calculated by:
\[ E_{spring} = \frac{1}{2} \cdot k \cdot x^2 \]
In this problem, this energy is released as kinetic energy when the block is freed from the spring. We calculated the work done by the spring, converting it to kinetic energy as the block pushed back out, allowing us to understand how the spring's stored energy became the driving force for the block's initial motion.
Kinetic Energy
Kinetic energy is the energy possessed by an object due to its motion. The formula to calculate the kinetic energy \( KE \) is:
\[ KE = \frac{1}{2} \cdot m \cdot v^2 \]
where \( m \) is the mass of the object and \( v \) is its velocity. As the block slides across the surface, its kinetic energy decreases until it is stopped by friction.
In the given solution, the initial kinetic energy originates from the energy released by the compressed spring. As the block moves and slows due to friction, its kinetic energy diminishes to zero. The work-energy theorem helped us connect the decrease in kinetic energy with the work done by friction, providing insight into the physical interactions stopping the block. Understanding \( KE \) is crucial in isolating the effects of different forces involved in motion.
\[ KE = \frac{1}{2} \cdot m \cdot v^2 \]
where \( m \) is the mass of the object and \( v \) is its velocity. As the block slides across the surface, its kinetic energy decreases until it is stopped by friction.
In the given solution, the initial kinetic energy originates from the energy released by the compressed spring. As the block moves and slows due to friction, its kinetic energy diminishes to zero. The work-energy theorem helped us connect the decrease in kinetic energy with the work done by friction, providing insight into the physical interactions stopping the block. Understanding \( KE \) is crucial in isolating the effects of different forces involved in motion.
