Two blocks of masses m 1 and m 2 ( m 1 > m 2 ) are placed on a frictionless tabl

Question

Two blocks of masses m1 and m2 (m1 > m2) are placed on a frictionless table in contact with each other. A horizontal force of magnitude F is applied to the block of mass m1 in the figure below.

(a) If P is the magnitude of the contact force between the blocks, draw the free-body diagrams for each block. (Do this on paper. Your instructor may ask you to turn in this work.)

(b) What is the net force on the system consisting of both blocks? (Use the following as necessary: P and F.)

(c) What is the net force acting on m1? (Use the following as necessary: P and F.)

(d) What is the net force acting on m2? (Use the following as necessary: P and F.)

(e) Write the x-component of Newton's second law for each block. (Do this on paper. Your instructor may ask you to turn in this work.)

(f) Solve the resulting system of two equations and two unknowns, expressing the acceleration a and contact force Pin terms of the masses and force. (Use the following as necessary: m1, m2, and F.)

(g) How would the answers change if the force had been applied to m2 instead? (Hint: use symmetry; don't calculate!) (Do this on paper. Your instructor may ask you to turn in this work.)

Is the contact force larger, smaller, or the same in this case?

largersmaller    the same

Why?

Fnet = Two blocks of masses m1 and m2 (m1 > m2) are placed on a frictionless table in contact with each other. A horizontal force of magnitude F is applied to the block of mass m1 in the figure below. (a) If P is the magnitude of the contact force between the blocks, draw the free-body diagrams for each block. (Do this on paper. Your instructor may ask you to turn in this work.) (b) What is the net force on the system consisting of both blocks? (Use the following as necessary: P and F.) (c) What is the net force acting on m1? (Use the following as necessary: P and F.) (d) What is the net force acting on m2? (Use the following as necessary: P and F.) (e) Write the x-component of Newton's second law for each block. (Do this on paper. Your instructor may ask you to turn in this work.) (f) Solve the resulting system of two equations and two unknowns, expressing the acceleration a and contact force Pin terms of the masses and force. (Use the following as necessary: m1, m2, and F.) (g) How would the answers change if the force had been applied to m2 instead? (Hint: use symmetry; don't calculate!) (Do this on paper. Your instructor may ask you to turn in this work.) Is the contact force larger, smaller, or the same in this case?

Explanation / Answer

Two blocks of masses m1 and m2 are placed on a table in contact with each other as shown in the figure below. The coefficient of kinetic friction between the block of mass m1 and the table is ?1, and that between the block of mass m2 and the table is ?2. A horizontal force of magnitude F is applied to the block of mass m1. We wish to find P, the magnitude of the contact force between the blocks.

(b) What is the net force on the system of two blocks? (Use any variable or symbol stated above along with the following as necessary: F, P, and g.)

Fnet = _F - (?1*m1+ ?2*m2)*g_ N

(c) What is the net force acting on m1? (Use any variable or symbol stated above along with the following as necessary: F, P, and g.)
Fnet,1= _F - (?1*m1)*g - P_ N

(d) What is the net force acting on m2? (Use any variable or symbol stated above along with the following as necessary: F, P, and g.)
Fnet,1=__F - (?2*m2)*g + P_ N ---------------------- (1)


(f) Solve the two equations in two unknowns for the acceleration a of the blocks in terms of the masses, the applied force F, the coefficients of friction, and g. (Use any variable or symbol stated above along with the following as necessary: F.)
a=_[{F/(m1+m2)} - {(?1*m1+ ?2*m2)/(m1+m2)}*g]_ m/s^2 -----------(2)

(g) Find the magnitude P of the contact force between the blocks in terms of the same quantities.
From (1) we get
P=_Fnet,1 - F + (?2*m2)*g _ N -------------------------------- (3)
From (2), we get
Fnet,1 = m1*[{F/(m1+m2)} - {(?1*m1+ ?2*m2)/(m1+m2)}*g] -----------(4) Substituting (4) in (3) we get
P = _m1*[{F/(m1+m2)} - {(?1*m1+ ?2*m2)/(m1+m2)}*g] - F+(?2*m2)*g_ or
P = _[-{m2/(m1+m2)}F+(?2*m2^2 - ?1*m1^2)*g]_ N

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