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A 1.00 -kg block initially at rest at the top of a 4.00 -m incline with a slope of \(45.0^{\circ}\) begins to slide down the incline. The upper half of the incline is frictionless, while the lower half is rough, with a coefficient of kinetic friction \(\mu_{k}=0.300\) a) How fast is the block moving midway along the incline, before entering the rough section? b) How fast is the block moving at the bottom of the incline?

Short Answer

Expert verified
Answer: The block is moving at a velocity of 5.28 m/s midway along the incline and 4.41 m/s at the bottom of the incline.

Step by step solution

01

Calculate gravitational potential energy at the top of the incline

First, we need to calculate the initial gravitational potential energy (PE) at the top of the incline. We will use the equation: \(PE = mgh\), where m is the mass of the block, g is the acceleration due to gravity (9.81 \(m/s^2\)), and h is the elevation from the horizontal at the top of the incline. To find h, we use the formula: \(h = L\sin{\theta}\), where L is the length of the incline, and \(\theta\) is the angle of inclination (45 degrees). h = 4.00 m * sin(45.0°) = 2.83 m PE = 1.00 kg * 9.81 \(m/s^2\) * 2.83 m = 27.74 J
02

Calculate the velocity of the block midway along the incline

Since the upper half of the incline is frictionless, we can use the conservation of mechanical energy to find the velocity of the block midway along the incline (at L/2). The equation for conservation of mechanical energy is: \(PE_i + KE_i = PE_f + KE_f\). Initially, at the top of the incline (\(PE_i\) = 27.74 J, \(KE_i\) = 0, since the block is at rest), and at the middle of the incline (\(PE_f\) = \(PE_i\)/2, \(KE_f\) = ?). PE_f = 27.74 J / 2 = 13.87 J Now we can solve for \(KE_f\): \(27.74 J = 13.87 J + KE_f\) KE_f = 13.87 J To find the velocity (v) of the block at the middle of the incline, we use the equation: \(KE = \frac{1}{2}mv^2\) v = \(\sqrt{\frac{2 * KE}{m}}\) = \(\sqrt{\frac{2 * 13.87J}{1 kg}}\) = 5.28 \(m/s\)
03

Calculate the work done by friction in the lower half of the incline

In the lower half of the incline with kinetic friction, we need to determine the work done by friction. The work-energy theorem states that the work done on an object to change its velocity is equal to the change in its kinetic energy. The work done by friction is given by \(W = -\mu_kFNd\), where \(\mu_k\) is the coefficient of kinetic friction (0.300), FN is the normal force, and d is the distance (lower half of the incline). The normal force FN equals m*g*cos(45.0°), so we can substitute and rewrite the work done equation as: \(W = -\mu_kmg\cos{\theta}d\) W = -0.300 * 1 kg * 9.81 \(m/s^2\) * cos(45.0°) * (4.00m / 2) = -4.137 J
04

Calculate the final velocity of the block at the bottom of the incline

Now, we can use the work-energy theorem to find the final velocity (v_f) of the block at the bottom of the incline: \(W = KE_f - KE_i\) \(-4.137 J = KE_f - 13.87 J\) \(KE_f\) = 9.73 J Finally, we use the kinetic energy equation to find the final velocity: \(v_f = \sqrt{\frac{2 * KE_f}{m}}\) = \(\sqrt{\frac{2 * 9.73 J}{1 kg}}\) = 4.41 \(m/s\) The answers are: a) The block is moving at a velocity of 5.28 \(m/s\) midway along the incline, before entering the rough section. b) The block is moving at a velocity of 4.41 \(m/s\) at the bottom of the incline.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gravitational Potential Energy
Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. Typically, it's the energy associated with the height of the object above a reference point, such as the ground. The basic formula for calculating GPE is \( PE = mgh \), where \( m \) represents mass, \( g \) stands for the acceleration due to gravity, and \( h \) is the height above the reference point.

In our physics problem, the block has GPE at the top of the incline due to its height. As the block begins sliding down, this potential energy is converted into kinetic energy. GPE is pivotal in understanding how energy is stored in the block before it starts moving, which will later convert into motion or work against forces like friction.
Conservation of Mechanical Energy
The conservation of mechanical energy principle is a fundamental concept in physics. It states that if no external work is done on a system, the total mechanical energy (sum of potential and kinetic energies) in the system remains constant. This principle is incredibly useful in solving problems involving the motion of objects where only conservative forces, like gravity, are doing work.

For the block sliding down an incline, we see the conservation of mechanical energy in action during its descent on the frictionless section. The initial potential energy at the top is wholly transformed into kinetic energy when it reaches the midway point before it hits the rough section with kinetic friction. The step-by-step solution demonstrates how you can use this principle to calculate the velocity of the block at the midway point by setting the total energy at the top equal to the total energy at the midpoint.
Kinetic Friction
Kinetic friction occurs when two surfaces move over each other and is described by the equation \( f_k = \mu_k F_N \), where \( f_k \) is the force of kinetic friction, \( \mu_k \) is the coefficient of kinetic friction, and \( F_N \) is the normal force. Coefficient \( \mu_k \) is a unitless value that represents the frictional properties of the two surfaces in contact.

In the context of our problem, once the block reaches the rough part of the incline, kinetic friction starts acting against its motion, doing negative work and thus removing energy from the system. This is crucial for determining the block’s final velocity at the bottom since we must account for the energy loss due to the work done by friction. The rougher the surface (higher \( \mu_k \) value), the more energy is lost, resulting in a slower block at the bottom.
Work-Energy Theorem
The work-energy theorem is an important tool in physics that relates work to the change in kinetic energy of an object. It states that the work done by all forces acting on a body will result in an equivalent change in the body’s kinetic energy. In equation form, \( W = \Delta KE \), where \( W \) is work and \( \Delta KE \) represents the change in kinetic energy.

This theorem is applied in the latter half of the problem to find the final velocity of the block at the bottom of the incline. The negative work done by kinetic friction, calculated in a previous step, reduces the block's kinetic energy. By quantifying this work, the theorem allows us to accurately calculate the resulting decrease in velocity due to friction.

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