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Chapter 6: Problem 35

Calculate the force \(F(y)\) associated with each of the following potential energies: a) \(U=a y^{3}-b y^{2}\) b) \(U=U_{0} \sin (c y)\)

Short Answer

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Question: Calculate the force, \(F(y)\), associated with the following potential energies: a) \(U = a y^{3} - b y^{2}\) b) \(U = U_{0} \sin (c y)\) Answer: a) \(F(y) = -3 a y^{2} + 2 b y\) b) \(F(y) = -U_{0} c \cos (c y)\)

Step by step solution

01

Calculate Force for Potential Energy a) \(U=a y^{3}-b y^{2}\)

First, we need to find the derivative of the potential energy: \(\frac{dU}{dy}\). So, differentiate the given potential energy \(U = a y^{3} - b y^{2}\) with respect to \(y\). \(\frac{dU}{dy} = \frac{d}{dy}(a y^{3} - b y^{2}) = 3 a y^{2} - 2 b y.\) To find the force, we simply multiply this result by \(-1\): \(F(y) = -\frac{dU}{dy} = -(3 a y^{2} - 2 b y) = -3 a y^{2} + 2 b y\).
02

Calculate Force for Potential Energy b) \(U=U_{0} \sin (c y)\)

Again, we need to find the derivative of the potential energy: \(\frac{dU}{dy}\). So, differentiate the given potential energy \(U = U_{0} \sin (c y)\) with respect to \(y\). \(\frac{dU}{dy} = \frac{d}{dy}(U_{0} \sin (c y)) = U_{0} c \cos (c y).\) To find the force, we multiply this result by \(-1\): \(F(y) = -\frac{dU}{dy} = -U_{0} c \cos (c y)\). The force associated with each potential energy is: a) \(F(y) = -3 a y^{2} + 2 b y\) b) \(F(y) = -U_{0} c \cos (c y)\)

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Most popular questions from this chapter

A spring with a spring constant of \(500 . \mathrm{N} / \mathrm{m}\) is used to propel a 0.500 -kg mass up an inclined plane. The spring is compressed \(30.0 \mathrm{~cm}\) from its equilibrium position and launches the mass from rest across a horizontal surface and onto the plane. The plane has a length of \(4.00 \mathrm{~m}\) and is inclined at \(30.0^{\circ} .\) Both the plane and the horizontal surface have a coefficient of kinetic friction with the mass of \(0.350 .\) When the spring is compressed, the mass is \(1.50 \mathrm{~m}\) from the bottom of the plane. a) What is the speed of the mass as it reaches the bottom of the plane? b) What is the speed of the mass as it reaches the top of the plane? c) What is the total work done by friction from the beginning to the end of the mass's motion?

a) If the gravitational potential energy of a 40.0 -kg rock is 500 . J relative to a value of zero on the ground, how high is the rock above the ground? b) If the rock were lifted to twice its original height, how would the value of its gravitational potential energy change?

A particle is moving along the \(x\) -axis subject to the potential energy function \(U(x)=1 / x+x^{2}+x-1\) a) Express the force felt by the particle as a function of \(x\). b) Plot this force and the potential energy function. c) Determine the net force on the particle at the coordinate \(x=2.00 \mathrm{~m}\)

A 1.00 -kg block is resting against a light, compressed spring at the bottom of a rough plane inclined at an angle of \(30.0^{\circ}\); the coefficient of kinetic friction between block and plane is \(\mu_{\mathrm{k}}=0.100 .\) Suppose the spring is compressed \(10.0 \mathrm{~cm}\) from its equilibrium length. The spring is then released, and the block separates from the spring and slides up the incline a distance of only \(2.00 \mathrm{~cm}\) beyond the spring's normal length before it stops. Determine a) the change in total mechanical energy of the system and b) the spring constant \(k\).

Can a unique potential energy function be identified with a particular conservative force?
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