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A rod AB of mass M and length L is lying on a horizontal frictionless surface. A particle of mass m travelling along the surface hits the end of the rod with a velocity vo in a direction perpendicular to AB. The collision is plastic. After the collision the particle comes to rest.(a) Find the ratio m/M. (b) A point P on the rod is at rest immediately after collision. Find the distance AP. (c) Find the linear speed of the point P at time piL/3vo , after the collision.?
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A rod AB of mass M and length L is lying on a horizontal frictionless ...
Solution:
(a) Ratio of masses
- In a plastic collision, the momentum is conserved and the kinetic energy is lost.
- Let v be the velocity of the rod after the collision.
- The momentum conservation equation is:
mv0 = (M+m)v
- The kinetic energy loss is:
(1/2)mv0^2 = (1/2)(M+m)v^2 + (1/2)(1/3)ML^2v^2
- Solving for m/M, we get:
m/M = 3(v0/v - 1)/(1 + 3(v0/v)^2)
(b) Distance AP
- Let x be the distance of point P from A.
- The velocity of point P after the collision is the same as the velocity of the rod, which is v.
- The angular momentum conservation equation about A is:
mv0L = Mvx(x-L/2) + (1/3)ML^2v
- Solving for x, we get:
x = L/2 - (v0L/3v)
(c) Linear speed of point P
- Let t be the time after collision.
- The angular velocity of the rod at time t is:
ω = (3v/L)t
- The linear speed of point P at time t is:
vP = ω(AP)
- Substituting the expression for AP from part (b), we get:
vP = (3v/2L)(v0L/3v - L/2)
- Simplifying, we get:
vP = (v0/2)(1 - v0/v)
(a) Ratio of masses
- In a plastic collision, the momentum is conserved and the kinetic energy is lost.
- Let v be the velocity of the rod after the collision.
- The momentum conservation equation is:
mv0 = (M+m)v
- The kinetic energy loss is:
(1/2)mv0^2 = (1/2)(M+m)v^2 + (1/2)(1/3)ML^2v^2
- Solving for m/M, we get:
m/M = 3(v0/v - 1)/(1 + 3(v0/v)^2)
(b) Distance AP
- Let x be the distance of point P from A.
- The velocity of point P after the collision is the same as the velocity of the rod, which is v.
- The angular momentum conservation equation about A is:
mv0L = Mvx(x-L/2) + (1/3)ML^2v
- Solving for x, we get:
x = L/2 - (v0L/3v)
(c) Linear speed of point P
- Let t be the time after collision.
- The angular velocity of the rod at time t is:
ω = (3v/L)t
- The linear speed of point P at time t is:
vP = ω(AP)
- Substituting the expression for AP from part (b), we get:
vP = (3v/2L)(v0L/3v - L/2)
- Simplifying, we get:
vP = (v0/2)(1 - v0/v)
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A rod AB of mass M and length L is lying on a horizontal frictionless surface. A particle of mass m travelling along the surface hits the end of the rod with a velocity vo in a direction perpendicular to AB. The collision is plastic. After the collision the particle comes to rest.(a) Find the ratio m/M. (b) A point P on the rod is at rest immediately after collision. Find the distance AP. (c) Find the linear speed of the point P at time piL/3vo , after the collision.?
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A rod AB of mass M and length L is lying on a horizontal frictionless surface. A particle of mass m travelling along the surface hits the end of the rod with a velocity vo in a direction perpendicular to AB. The collision is plastic. After the collision the particle comes to rest.(a) Find the ratio m/M. (b) A point P on the rod is at rest immediately after collision. Find the distance AP. (c) Find the linear speed of the point P at time piL/3vo , after the collision.? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A rod AB of mass M and length L is lying on a horizontal frictionless surface. A particle of mass m travelling along the surface hits the end of the rod with a velocity vo in a direction perpendicular to AB. The collision is plastic. After the collision the particle comes to rest.(a) Find the ratio m/M. (b) A point P on the rod is at rest immediately after collision. Find the distance AP. (c) Find the linear speed of the point P at time piL/3vo , after the collision.? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A rod AB of mass M and length L is lying on a horizontal frictionless surface. A particle of mass m travelling along the surface hits the end of the rod with a velocity vo in a direction perpendicular to AB. The collision is plastic. After the collision the particle comes to rest.(a) Find the ratio m/M. (b) A point P on the rod is at rest immediately after collision. Find the distance AP. (c) Find the linear speed of the point P at time piL/3vo , after the collision.?.
A rod AB of mass M and length L is lying on a horizontal frictionless surface. A particle of mass m travelling along the surface hits the end of the rod with a velocity vo in a direction perpendicular to AB. The collision is plastic. After the collision the particle comes to rest.(a) Find the ratio m/M. (b) A point P on the rod is at rest immediately after collision. Find the distance AP. (c) Find the linear speed of the point P at time piL/3vo , after the collision.? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A rod AB of mass M and length L is lying on a horizontal frictionless surface. A particle of mass m travelling along the surface hits the end of the rod with a velocity vo in a direction perpendicular to AB. The collision is plastic. After the collision the particle comes to rest.(a) Find the ratio m/M. (b) A point P on the rod is at rest immediately after collision. Find the distance AP. (c) Find the linear speed of the point P at time piL/3vo , after the collision.? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A rod AB of mass M and length L is lying on a horizontal frictionless surface. A particle of mass m travelling along the surface hits the end of the rod with a velocity vo in a direction perpendicular to AB. The collision is plastic. After the collision the particle comes to rest.(a) Find the ratio m/M. (b) A point P on the rod is at rest immediately after collision. Find the distance AP. (c) Find the linear speed of the point P at time piL/3vo , after the collision.?.
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