If the initial energy of the system was 845.54 J and the final energy of the system is -104.99 J , how much work has been done?
2 answers:
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The spring-mass system moves by simple harmonic motion, where there is a continuous conversion from elastic potential energy to kinetic energy and viceversa.
The total mechanical energy of the system at any moment of the motion is
where the first term U is the elastic potential energy, with k being the spring constant and x the displacement of the spring with respect to its rest position, and the second term K is the kinetic energy, with m being the mass of the object attached to the spring and v its speed.
The total energy E is constant during the oscillation of the spring, but the values of U and K change. In fact, when the displacement of the spring is maximum (x is maximum), then all the energy is potential energy U, because the speed of the object is zero (it's the moment when the mass is changing direction). On the contrary, when the mass crosses the equilibrium position (rest position) of the spring, then the potential energy is zero (U=0) because the displacement is zero (x=0), and so all the energy is kinetic energy of the motion, and so K is maximum.
The total mechanical energy of the system at any moment of the motion is
where the first term U is the elastic potential energy, with k being the spring constant and x the displacement of the spring with respect to its rest position, and the second term K is the kinetic energy, with m being the mass of the object attached to the spring and v its speed.
The total energy E is constant during the oscillation of the spring, but the values of U and K change. In fact, when the displacement of the spring is maximum (x is maximum), then all the energy is potential energy U, because the speed of the object is zero (it's the moment when the mass is changing direction). On the contrary, when the mass crosses the equilibrium position (rest position) of the spring, then the potential energy is zero (U=0) because the displacement is zero (x=0), and so all the energy is kinetic energy of the motion, and so K is maximum.
Answer:
The coefficient of friction is (F/(19.6·m)
Explanation:
The given parameters are;
The force applied to the immovable body = F
The time duration the force acts = t
The time the body spends in motion = 3·t
The acceleration due to gravity, g = 9.8 m/s²
From Newton's second law of motion, we have;
The impulse of the force = F × t = m × Δv₁
Where;
Δv₁ = v₁ - 0 = v₁
The impulse applied by the force of friction, is
× (3·t - t) =
× (2·t)
Given that the motion of the object is stopped by the frictional force, we have;
The impulse due to the frictional force = Momentum change = m × Δv₂ = × (2·t)
Where;
Δv₂ = v₂ - 0 = v₂
Given that the velocity, v₂, at the start of the deceleration = The velocity at the point the force ceased to act, v₁, we have;
m × Δv₂ = × (2·t) = m × Δv₁ = F × t
∴ × (2·t) = F × t
= F × t/(2·t) = F/2
The coefficient of dynamic friction, = Frictional force/(The weight of the body) = (F/2)/(9.8 × m)
= (F/(19.6·m)
The coefficient of friction, = (F/(19.6·m)
The graphs of the acceleration as a function of mass and of the acceleration as a function of force are in attachment.
Explanation:
To answer both parts of the question, we refer to Newton's second law, which states that:
where
F is the net force on an object
m is the mass of the object
a is its acceleration
a)
To answer this part, we re-arrange the previous equation as follows:
Therefore, we notice that if a constant force is applied, the acceleration is inversely proportional to the mass of the object:
this means that if the mass increases, the acceleration decreases, and if the mass decreases, the acceleration increases.
In a graph of acceleration vs mass, the curve representing this relationship would be a hyperbole. The graph is shown as the first graph in the attached picture.
b)
As before, we re-arrange the previous equation as follows:
Here we notice that if the object has a constant mass, the acceleration is directly proportional to the force applied on the object:
this means that when the force increases, the acceleration increases, and when the force decreases, the acceleration decreases.
In a graph of acceleration vs force, the curve representing this relationship would be a straight line, as shown in the second graph in the attached picture.
Learn more about Newton's second law of motion here:
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