A triangular rod of length L and mass M has a nonuniform linear mass density given by the equation λ=γx2, where γ=3M/(L3) and x is the distance from point P at the left end of the rod. (a) Using integral calculus, show that the rotational inertia I of the rod about an axis perpendicular to the page and through point PP is (3/5)ML2.

A triangular rod of length L and mass M has a nonuniform linear mass density given by the equation λ=γx2, where γ=3M/(L3) and x is the distance from point P at the left end of the rod. (a) Using integral calculus, show that the rotational inertia I of the rod about an axis perpendicular to the page and through point PP is (3/5)ML2.

Related questions

Q: Consider a flat disk of radius 0.3 m. The mass per area of the disk is a function of r, the…

Q: The figure shows three 0.0105 kg particles that have been glued to a rod of length L=6.50 cm and…

Q: The ammonia molecule can be treated as a symmetric rigid rotator, with its 3 hydrogen atoms lying in…

A: The ammonia molecule can be treated as a symmetric rigid rotator, with its 3 hydrogen atoms lying in…

Q: rim makes an angle of 57.3°with the horizontal at this time. At t = 2.00 s, find the following. (a)…

Q: A solid and uniform disk of mass 8.0 kg and radius 25 cm is rotating at 1506 revolutions per minute…

Q: A light, rigid rod of length = 1.00 m joins two particles, with masses m₁ = 4.00 kg and m₂ = 3.00…

Q: A torque wrench grips a bolt at the origin of coordinates, with the free end of the torque wrench…

Q: For this question consider a cylindrical shell that has a radius R = 4.6 m, and a mass M = 4.35 kg.…

Q: In the figure, two particles, each with mass m = 0.80 kg, are fastened to each other, and to a…

A: mass of particles , m = 0.8 kg mass of rod , M = 1.1 kg Angular speed , ω = 0.34 rad/s d = 5.4 cm =…

Q: The figure shows three 0.01 kg particles that have been glued to a rod of length L=6 cm and…

Q: include units

A: a)The rotational inertia is 0.0248267kgm2b)The kinetic rotational energy is…

Q: A uniform rod of mass 315 g and length 50.0 cm rotates in a horizontal plane about a fixed,…

A: Given data:The mass of rod is M= 315 g.The length of rod is l=50 cm.The mass of beads is m g.The…

Q: A thin uniform rod of mass M and length L is nailed to the tabletop in an experiment on rotation. If…

Q: A uniform circular wheel (I=M R?) with radius, R=1.0 m and mass, M=50.0 kg is spinning at 120…

Q: (a) Starting from I = r² dm show that a thin rod, length L, mass M, spinning about an axis at one…

Q: A uniform solid disk has mass M and radius R. A child of mass m = M/2 is standing at two-thirds of…

A: The objective of the question is to find the rotational inertia (moment of inertia) of the system…

Q: A circular cylinder is being rotate as shown in the figure below with an axis of rotation going…

A: Given value--- h = 17 cm. r = 3.1 cm. d = 16 cm. density = 7276 kg/m3. We have to find--- What…

Q: Four identical particle of mass 0.5 kg each are placed at the vertices of a 2 m x 2m square and held…

A: thank you

Q: A uniform stick of mass M=2.98kg and length L=1.5m which is suspended horizontally with end B at the…

Q: A large wooden wheel (mass 2 kg, radius 0.5 m) is mounted on an axle so as to rotate freely about…

A: A large wooden wheel (mass 2 kg, radius 0.5 m) is mounted on an axle so as to rotate freely about…

Q: The figure shows three 0.0133 kg particles that have been glued to a rod of length L=6.23 cm and…

Q: A wheel 2.10 m in diameter lies in a vertical plane and rotates about its central axis with a…

Q: A hollow sphere of mass M and radius R = 0.15 m, with rotational inertia I = 0.040 kg · m2 about a…

A: Concept: According to the Hint given, The moment of inertia of the sphere is I=23MR2 here, M= mass…

Q: Suppose you have a long, thin rod of length 1.51m that is free to rotate about a fixed end. If the…

Question

A triangular rod of length L and mass M has a nonuniform linear mass density given by the equation λ=γx², where γ=3M/(L³) and x is the distance from point P at the left end of the rod.

(a) Using integral calculus, show that the rotational inertia I of the rod about an axis perpendicular to the page and through point PP is (3/5)ML².

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Similar questions

The figure shows three 0.0186 kg particles that have been glued to a rod of length L=5.67 cm and negligible mass. The assembly can rotate around a perpendicular axis through point O at the left end. If we remove one particle (that is, 33% of the mass), by what percentage does the rotational inertia of the assembly around the rotation axis decrease when that removed particle is (a) the innermost one and (b) the outermost one? Axis (a) Number Units (b) Number Units
Solid spheres have a rotational inertia about an axis through their center of mass given by I = 2/5 M R2, where M is the mass of the sphere and R is the radius of the sphere. Consider a particular sphere of radius 1.2 m made out of a material of density 300 kg/m3. Using the parallel axis theorem, calculate the rotational inertia of the sphere about a parallel axis shifted 0.33 m away from the center. (Please answer to the fourth decimal place - i.e 14.3225)
A hollow sphere of mass M and radius R = 0.15 m, with rotational inertia I = 0.040 kg · m² about a line through its center of mass, rolls without slipping up a surface inclined at 0 = 30° to the horizontal. At a certain position, say at a height hi from the ground, the sphere's total kinetic energy (its center of mass kinetic energy plus its rotational kinetic energy) is 20 J. (Hint: The moment of inertia of such a halo sphere is I = MR². The translational speed of a point on the surface of the sphere is the same as the speed of the center of mass of the sphere) (a) What ratio of this total kinetic energy is rotational? (b) What is the speed of the center of mass of the sphere at this position? (5 points) 1 m up the incline from this (initial) position, using When the sphere has moved d conservation of energy, find (c) its total kinetic energy ), and (d) the speed of its center of mass
In the figure below, the disk, of mass 440 g and radius 3.5 cm, is rotating at 180 rpm on a frictionless shaft of negligible radius. The upper disk, of mass 270 g and radius 2.3 cm, is initially not rotating. It drops freely down onto the lower dis, and frictional forces bring the two disk to a common rotational speed. Find (a) that common speed and (b) the fraction of the initial kinetic energy lost to friction.
The figure shows three 0.0146 kg particles that have been glued to a rod of length L-5.58 cm and negligible mass. The assembly can rotate around a perpendicular axis through point O at the left end. If we remove one particle (that is, 33% of the mass), by what percentage does the rotational inertia of the assembly around the rotation axis decrease when that removed particle is (a) the innermost one and (b) the outermost one? (a) Number (b) Number Axis Units Units 0 m m m |a+d+d+
Engineering Dynamics need help from 4,5,6,7 thank you A ball of mass m is moving along a vertical semi-cylinder of radius R as it is guided by the arm OA. The arm moves in a clockwise direction with a constant angular velocity ω. Assume 0° ≤ Φ ≤ 90°. Neglect any friction. Neglect also the size of the ball and the thickness of the arm. Find the relationship between r, R and θ where r is the distance between O and the ball. Draw a free body diagram of the ball assuming that it is in contact with the cylinder and the arm OA. Write the equations of motion in the (r, θ) coordinate system. Find the normal force acting on the ball by the cylinder for Φ = Φ0. Find the normal force acting on the ball by the bar for Φ = Φ0. Determine the angle Φ at which the ball loses contact with the cylinder. Take m = 1 kg, R = 1.4 m, ω = 0.5 rad/s, and Φ = 60°
The system is made up of the 1-m, 7.1-kg uniform rod AB and the two uniform disks (each of 0.17 m radius and 1.9 kg mass). If the system is released from rest when = 63°, determine the total external work done to this system when rod AB has just become horizontal. Assume the disks roll without slipping. Please pay attention: the numbers may change since they are randomized. Your answer must include 2 places after the decimal point, and proper Sl unit. Take g = 9.81 m/s². B m Ө Your Answer: Answer units B
The figure shows an arrangement of 15 identical disks that have been glued together in a rod-like shape of length L = 1.4600 m and (total) mass M = 160.0 mg. The arrangement can rotate about a perpendicular axis through its central disk at point O. (a) What is the rotational inertia of the arrangement about that axis? Give your answer to four significant figures. (b) If we approximated the arrangement as being a uniform rod of mass M and length L, what percentage error would we make in using the formula in Table 10-2e to calculate the rotational inertia? ooooooooooo (a) Number i (b) Number i Units Units
The figure shows three 0.0114 kg particles that have been glued to a rod of length L= 5.70 cm and negligible mass. The assembly can rotate around a perpendicular axis through point O at the left end. If we remove one particle (that is, 33% of the mass), by what percentage does the rotational inertia of the assembly around the rotation axis decrease when that removed particle is (a) the innermost one and (b) the outermost one? Axis m m (a) Number i Units (b) Number i Units >