A mass M is initially at rest on a horizontal surface, μs=0.30 and μk=0.20 . A horizontal string then pulls M with a tension T. N is the normal component of the contact force. All quantities below are magnitudes. Answers are either true or false.

1)The total force on M is μkN if M remains at rest.
2)If M does not accelerate, then T ≤ μsN
3)M will accelerate if T exceeds μkN
4)T equals μkN if M remains at rest.
5)The value of N equals Mg
6)M will accelerate if T exceed μsN

To determine whether the given statements are true or false, we need to analyze the forces acting on the mass M.

1) The total force on M is μkN if M remains at rest.
False. According to Newton's First Law of Motion, an object at rest remains at rest unless acted upon by an external force. In this case, if M remains at rest, it means that there is no net force acting on it. The frictional force, given by μkN, opposes the motion and would need to be balanced by an equal and opposite external force. Therefore, the total force on M would be zero, not μkN.

2) If M does not accelerate, then T ≤ μsN.
True. If M does not accelerate, it means that the net force acting on it is zero. The maximum static frictional force is given by μsN. So, in order for M not to accelerate, the tension T must not exceed the maximum static frictional force, i.e., T ≤ μsN.

3) M will accelerate if T exceeds μkN.
True. The frictional force acting on the mass M is given by μkN. If the tension T exceeds this frictional force, there will be a net force acting on M in the direction of T. As a result, M will accelerate.

4) T equals μkN if M remains at rest.
False. As explained in statement 1, if M remains at rest, then the total force on it is zero. Therefore, the tension T would need to be equal to the maximum static frictional force (μsN) to balance the opposing frictional force.

5) The value of N equals Mg.
True. The normal force N is the force exerted by the horizontal surface on the mass M in the upward direction. In this case, as the mass is at rest on the horizontal surface, the normal force N would be equal to the weight of the object, which is given by Mg.

6) M will accelerate if T exceeds μsN.
False. If T exceeds the maximum static frictional force (μsN), the object will not start accelerating immediately. Instead, the static frictional force will match the applied force until the maximum is reached. Only when T exceeds the maximum static frictional force can M start to accelerate.