# Force Problems

On this page I put together a collection of force problems to help you understand forces better. The required equations and background reading to solve these problems are given on the

friction page, the

equilibrium page, and

Newton's second law page.

**Problem # 1**
A ball of mass

*m* is hanging on a wall with a string, as shown. The string makes an angle

*θ* with the wall. What is the tension in the string?

__Answer:__ *mg*/cos

*θ*
**Problem # 2**
A block of mass

*m* is hanging off the ceiling using three strings, as shown. What is the tension in the three strings?

__Answer:__ Bottom string tension is

*mg*, right string tension is

*mg*/(sin

*α*+cos

*α*tan

*θ*), left string tension is

*mg*/(sin

*θ*+cos

*θ*tan

*α*)

**Problem # 3**
A block of mass

*M* is being pulled at constant velocity on a horizontal floor with a force

*F*, at angle

*θ* as shown. The coefficient of kinetic friction is

*μ*_{k}, between block and floor. Determine the force

*F*.

__Answer:__ *F* =

*Mg**μ*_{k}/(cos

*θ*+

*μ*_{k}sin

*θ*)

**Problem # 4**
A block of mass

*M* is being pushed at constant velocity on a horizontal floor with a force

*F*, at angle

*θ* as shown. The coefficient of kinetic friction is

*μ*_{k}, between block and floor. Determine the force

*F*.

__Answer:__ *F* =

*Mg**μ*_{k}/(cos

*θ*—

*μ*_{k}sin

*θ*)

**Problem # 5**
A block of mass

*M* is being pulled at constant velocity on a horizontal floor with a force

*F*. The coefficient of kinetic friction is

*μ*_{k}, between block and floor. Determine the force

*F*.

__Answer:__ *F* =

*Mg**μ*_{k}
**Problem # 6**
A block is lying on a horizontal floor. The weight of the block is

*W*. What is the normal force

*N* between the block and floor?

__Answer:__ *N* =

*W*
**Problem # 7**
A block of mass

*M* is sitting on a surface inclined at angle

*θ*. Given that the coefficient of static friction is

*μ*_{s} between block and surface, what is the maximum angle

*θ* so that the block doesn't slide?

Hint and answer
**Problem # 8**
Two blocks of mass

*M* and

*m* are sitting on a surface inclined at angle

*θ*. The blocks are touching. The coefficient of static friction between the smaller block and the surface is

*μ*_{s1}, and the coefficient of static friction between the larger block and the surface is

*μ*_{s2}. Formulate a mathematical inequality for the condition that no sliding occurs. There may be more than one inequality.

Hint and answer
**Problem # 9**
Two blocks of mass

*M* and

*m* are sitting on a horizontal floor. The blocks are touching. The coefficient of static friction between the smaller block and the floor is

*μ*_{s1}, and the coefficient of static friction between the larger block and the floor is

*μ*_{s2}. What is the minimum push force

*F* so that the blocks begin to move?

__Answer:__ *F*_{min} =

*Mg**μ*_{s2}+

*mg**μ*_{s1}
**Problem # 10**
A block of mass

*m* is sitting on a block of mass

*M*. The bottom block is sitting on a horizontal floor. The coefficient of static friction between the blocks is

*μ*_{s1}, and the coefficient of static friction between the bottom block and the floor is

*μ*_{s2}. What is the minimum pull force

*F* on the bottom block so that the blocks begin to move? Given that the coefficient of kinetic friction between the bottom block and the floor is

*μ*_{k}, what is the maximum pull force

*F* so that there is no slipping between the blocks?

Hint and answer
The hints and answers for these force problems will be given next.

**Hints And Answers For Force Problems**
__Hint and answer for Problem # 7__
The force of gravity pulling down on the block is

*F*_{1} =

*Mg*sin

*θ*. The maximum friction force opposing the sliding is

*F*_{2} =

*Mg*cos

*θ**μ*_{s}. At some angle

*θ* the block will be on the verge of sliding. This is the maximum angle

*θ* and occurs when

*F*_{1} =

*F*_{2}. The maximum angle

*θ* can be solved from this.

__Answer:__ *θ*_{max} = atan(

*μ*_{s})

__Hint and answer for Problem # 8__
There are two cases to consider:

*Case 1:* If

*μ*_{s1} ≤

*μ*_{s2} then the limiting case for no sliding occurs when

*θ* =

*θ*_{max} (where

*θ*_{max} = atan(

*μ*_{s1}), using the solution in Problem # 7). Therefore, for

*μ*_{s1} ≤

*μ*_{s2},

*θ* ≤ atan(

*μ*_{s1}), for no sliding.

*Case 2:* If

*μ*_{s1} >

*μ*_{s2} then sliding occurs only if the blocks slide down together, with the larger block pushing against the smaller block in front. The limiting case is when this is on the brink of happening, and the static friction is at its limit for both blocks. Since the blocks move together we can analyze them as one unit. The equation for static equilibrium along the plane of the incline is: (

*M*+

*m*)

*g*sin

*θ*—

*Mg*cos

*θ**μ*_{s2}—

*mg*cos

*θ**μ*_{s1} = 0. From this equation we can solve for

*θ*, and this is the maximum angle

*θ* for

*μ*_{s1} >

*μ*_{s2}, for which there is no sliding.

__Answer:__ For

*μ*_{s1} ≤

*μ*_{s2},

*θ* ≤ atan(

*μ*_{s1}). For

*μ*_{s1} >

*μ*_{s2},

*θ* ≤ atan{(

*M**μ*_{s2}+

*m**μ*_{s1})/(

*M*+

*m*)}

__Hint and answer for Problem # 10__
The blocks begin sliding when the force

*F* is just large enough to overcome the static friction between the bottom block and floor. This force is the minimum pull force.

The maximum pull force corresponds to the maximum acceleration that the blocks can experience without the top block slipping off the bottom block. By Newton's second law, the force exerted on the top block due to contact with the bottom block is

*F*_{top} =

*ma*, where

*a* is the acceleration of the blocks. Applying Newton's second law to the bottom block we have:

*F*—(

*M*+

*m*)

*g**μ*_{k}—

*F*_{top} =

*Ma*. To avoid slipping, the maximum value of

*F*_{top} is

*F*_{top} =

*mg**μ*_{s1}. This corresponds to the maximum acceleration and hence the maximum pull force. Combine the above three equations to solve for the maximum pull force.

__Answer:__ *F*_{min} = (

*M*+

*m*)

*g**μ*_{s2},

*F*_{max} = (

*M*+

*m*)

*g**μ*_{k}+

*mg**μ*_{s1}(1+

*M*/

*m*)

**Bonus Problems Related to Forces**
1.

*Is it possible to lift a car using fire hoses?*
2.

*Can a sailboat stranded in calm water start moving by blowing air into its sail with an onboard fan?*
I created a physics analysis for these two problems, in PDF format. It's available as an instant download through PayPal. Get it for $0.99 USD.

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