A string is wrapped tightly around the circumference of a wheel with a mass of 1

Question

A string is wrapped tightly around the circumference of a wheel with a mass of 1.23 kg and a

radius of 0.324 m. The string is attached to a block with a mass of 2.48 kg which is 0.34 m above the floor.

When the mass is released the wheel rotates freely as the string unwinds without slipping.

a. What is the initial energy of the system?

b. What is the moment of inertia of the wheel?

c. Write the expression for the total energy of the system just before the block hits the floor.

d. What is the velocity of the block just before it hits the floor?

e. What is the angular velocity of the wheel just before the block hits the floor?

Explanation / Answer

a) Initial energy of the system is the gravitational potential energy which is stored in mass block with respect to floor. Therefore, total initial energy of the system=(mass of block)*(acceleration due to gravity)*(height of block with respect to floor) = 2.48*9.81*0.34=8.271 joule b) moment of inertia of wheel which is also a disc , I = 1/2(m)*(r^2)=0.5*1.23*(0.324)^2=0.0645 kg*m^2 c) Just before the block hits the ground , wheel must be having rotational energy, block must be having kinetic energy so the expression of total energy = 1/2*(mass of block)*(velocity of block at that moment)^2 + 1/2*(moment of inertia of wheel)*(radius of wheel)^2=1/2*m*v^2 + 1/2*I*w^2 d) tension in the string will be = mass of block*g=2.48*9.81=24.329 N torque of this force on the wheel = 24.329*radius of wheel= 24.329*0.324=7.883 N-m its initial angular velocity = 0 , say its final angular velocity = w , and say at that moment velocity of block = u, as we know that u=r*w. total energy of system will remain same because there are no loss. Therefore 1/2(I)(w*w)+1/2(m)(u*u)= 8.271, where u =r*w. Substitute its value , Therefore, 1/2(I)(w*w) + 1/2(m)(r*w)^2=8.271 where = 0.0645 kg*m^2, r=0.324 m, m =2.48 kg the only unknown is w. 1/2(0.0645)(w)^2 +1/2(2.48)*(0.324)^2*(w)^2=8.271 0.162(w)^2=8.271 w=7.145 rad/sec velocity of block just before it hits the ground is = r*w=7.145*0.324=2.315 m/sec.

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