Question

6. A slender rod with length L has a mass per unit length that varies with distance from the left end, where x = 0, according to dm/dx = ?x, where ? has units of kg/m". a) Calculate the total mass M of the rod in terms of ? and L. b) Use the equation I = ? r"dm to calculate the moment of inertia of the rod for an axis at the left end, perpendicular to the rod. Express your result in terms of M and L. (Hint: Revise the problem- solving section 9) c) Assume that the rod with length L = 0.4 m and mass M = 3.0 kg is rotating about the axis as in part b. The angle through which the rod has turned varies with time according to ?(t) = (1.1 rad/s)t + (6.3 rad/s")t". What is the kinetic energy of the rod when it has rotated 0.1 rev? (Note: 1 rev = 2? rad)

          6. A slender rod with length L has a mass per unit length that varies with distance from the left
end, where x = 0, according to dm/dx = ?x, where ? has units of kg/m".
a) Calculate the total mass M of the rod in terms of ? and L.
b) Use the equation I = ? r"dm to calculate the moment of inertia of the rod for an axis at the left
end, perpendicular to the rod. Express your result in terms of M and L. (Hint: Revise the problem-
solving section 9)
c) Assume that the rod with length L = 0.4 m and mass M = 3.0 kg is rotating about the axis as
in part b. The angle through which the rod has turned varies with time according to
?(t) = (1.1 rad/s)t + (6.3 rad/s")t". What is the kinetic energy of the rod when it has
rotated 0.1 rev? (Note: 1 rev = 2? rad)

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6. A slender rod with length L has a mass per unit length that varies with distance from the left
end, where x = 0, according to dm/dx = ?x, where ? has units of kg/m".
a) Calculate the total mass M of the rod in terms of ? and L.
b) Use the equation I = ? r"dm to calculate the moment of inertia of the rod for an axis at the left
end, perpendicular to the rod. Express your result in terms of M and L. (Hint: Revise the problem-
solving section 9)
c) Assume that the rod with length L = 0.4 m and mass M = 3.0 kg is rotating about the axis as
in part b. The angle through which the rod has turned varies with time according to
?(t) = (1.1 rad/s)t + (6.3 rad/s")t". What is the kinetic energy of the rod when it has
rotated 0.1 rev? (Note: 1 rev = 2? rad)

Added by Michael Q.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Jacob Fry

10/07/2022

Step 1

Given that dm/dx = YX, we need to find the total mass M of the rod. We can do this by integrating dm/dx with respect to x from 0 to L: \[M = \int_{0}^{L} \frac{dm}{dx} dx = \int_{0}^{L} Yx dx = \frac{1}{2} YL^2\] Show more…

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6. A slender rod with length L has a mass per unit length that varies with distance from the left end, where x = 0, according to dm/dx = ̳x, where ̳ has units of kg/m". a) Calculate the total mass M of the rod in terms of ̳ and L. b) Use the equation I = ∫ r"dm to calculate the moment of inertia of the rod for an axis at the left end, perpendicular to the rod. Express your result in terms of M and L. (Hint: Revise the problem-solving section 9) c) Assume that the rod with length L = 0.4 m and mass M = 3.0 kg is rotating about the axis as in part b. The angle through which the rod has turned varies with time according to ̸(t) = (1.1 rad/s)t + (6.3 rad/s")t". What is the kinetic energy of the rod when it has rotated 0.1 rev? (Note: 1 rev = 2̰ rad)

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Transcript

00:01 So in this question, we're told that we have a slender rod of length l, whose mass per unit length varies.

00:13 So this is x equals zero.

00:16 This is x equals l.

00:18 And we have dm by dx equals gamma x.

00:24 So now the total mass, m, is going to be the integral dm by dx, dx, from zero to l.

00:35 So this is gamma times the integral from 0 to l of x, dx, which is a half gamma l squared.

00:44 So that's the total mass.

00:48 So now i equals the integral of r squared dm from the, so this is from the left end.

01:03 So this is going to be the integral of x squared, dm by d x, dx, dx from zero up to l.

01:15 So this is gamma times the integral from zero to l of x cubed dx.

01:21 So this is gamma, so this is going to be a quarter gamma l to the power of four.

01:29 But we want to express this in terms of m and l.

01:32 So this is going to be a half times a half gamma l squared times l squared.

01:40 So this is a half m l squared.

01:45 Now for part c, we have l equals 0 .4 meters, m equals 3 .0 kilograms.

01:56 And it's rotating about the axis as in part b.

02:00 So it's going to have i equals a half m l squared.

02:07 Now the angle, theta of t, equals 1 .1 radiance per second, times t plus 6 .3 radians per second squared times t squared.

02:27 So the kinetic energy, t, is a half i omega squared.

02:37 And omega is the angular velocity, so it's the rate of change of theta.

02:43 So this is a half i times, and we want this when it has rotated 0 .1 revolutions...

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