Question

Hand In Problem 2 A solid globe of mass M and radius R can rotate about its axis. A block of mass m is attached by a massless-string/pulley system as shown. As the block falls it causes the globe to spin. If the block starts at rest, at what speed does the block hit the ground after it falls a distance h? ' State your answer in terms of the given variables: M, R, m, h and g. Solve this problem using two different methods.v Method 1:v Use the concepts of forces, torques and accelerations to solve the problem. Make sure that you clearly use each step of the Problem Solving Strategies that we developed last term in Ph 211 along with the "twists" we add this term. Problem solving steps: a) Draw a picture! Include a "good" coordinate system. b) Draw a FBD c) If a body is being "torqued", draw a Torque Diagram. d) See if you can use a conservation law to make your life easier. f) Create an equation using Fnet- manet for every body in the problem- g) Create an equation using Tnet Imet for every torqued body in the problem. h) Always check to see if you have constant accelerations or changing accelerations. Method 2:* Use energy concepts to solve the problem. Note that in our problem solving steps above step d can be used in this case. Here's what to do if you have a problem that can be solved using a conservation law:* d) See if you can use a conservation law to make your life easier. +' i) Ifyou can use a conservation law, draw a before and an after picture.* ii) If there is a collision, then you probably want to use momentum/impulse concepts. Recall that momentum is a vector. Clearly show where the momentum is in the before picture for each dimension and where it is in the after picture for each dimension. Set the before-momentum for each dimension equal to the after-momentum of the same dimension. iii) If you use conservation of energy, clearly show where the energy is in the before picture and where the energy is in the after picture and set the two total energies equal to each other. Equations associated with the second part of Chapter 12. Continuing with the theme of the first part of this chapter, many translational concepts have rotational analogs:* KErotationalー½lo2+'

Answer 1

method 1:

from the free body diagram of the block, force equation is given as

mg - T = ma

T = mg - ma eq-1

Torque equation is given as

T R = I $\alpha$

using eq-1

(mg - ma) R = (0.4) M R² (a/R)

(mg - ma) = (0.4) Ma

a = mg/(m + (0.4)M)

v_i = initial speed = 0

Y = vertical distance travelled = h

V_f = final speed = ?

using the equation

V_f² = V_i² + 2 a Y

V_f² = 0² + 2 (mgh/(m + (0.4)M))

V_f = sqrt(2mgh/(m + (0.4)M))

Method 2:

using conservation of energy

initial potential energy = kinetic energy + rotational kinetic energy

mgh = (0.5) m v² + (0.5) I w²

mgh = (0.5) m v² + (0.5) (0.4) MR² (v/R)²

mgh = (0.5) m v² + (0.2) M v²

2mgh = (m + (0.4)M) v²

v = sqrt(2mgh/(m + (0.4)M))