A box of mass M sits on a table and is connected by a rope passing over a smooth support to another box of mass m that hangs over the edge of the table.
(a) What is the acceleration of M ignoring friction
(b) and if the coefficient of kinetic friction between box and table is
µ=0.345
a) mg = (M+m)a
b) mg - mu Mg = (M+m)a
To find the acceleration of the box (mass M) on the table, we need to analyze the forces acting on the system. Let's break it down step by step:
(a) Ignoring friction:
1. Draw a free body diagram for each box separately.
For the box hanging off the table (mass m), there are two forces acting on it: its gravitational force (mg) pulling it downward and the tension force (T) from the rope pulling it upward.
For the box on the table (mass M), there are three forces acting on it: its gravitational force (Mg) pulling it downward, the tension force (T) from the rope pulling it upward, and the normal force (N) from the table pushing it upward.
2. Apply Newton's second law (ΣF = ma) separately to each box.
For the hanging box (mass m):
- ΣFvertical = mg - T = masubscriptm (vertical motion)
- Therefore, T = mg - masubscriptm
For the box on the table (mass M):
- ΣFvertical = N - Mg - T = Ma (vertical motion)
- Therefore, N = Mg + T - Ma
3. Since the rope is assumed to be massless and does not stretch, the tension force (T) is the same for both boxes.
4. Substitute T from the equation for the hanging box into the equation for the box on the table.
- N = Mg + (mg - masubscriptm) - Ma
5. Rearrange the equation to isolate the acceleration term (a).
- N - Mg - mg + masubscriptm = Ma
6. Solving for a, we get the acceleration of the box (mass M) on the table.
- a = (N - Mg - mg + masubscriptm) / M
(b) Considering kinetic friction:
If there is a coefficient of kinetic friction (µ) between the box (mass M) and the table, we need to account for the frictional force (f) in the equation.
1. The frictional force (f) can be calculated using the equation f = µN, where N is the normal force.
2. Modify the equation for the box on the table to include the frictional force.
- N - Mg - mg + masubscriptm - µN = Ma
3. Rewrite the equation to isolate the acceleration term (a).
- a = (N - Mg - mg + masubscriptm - µN) / M
4. Now, plug in the values of the masses (M and m), gravitational acceleration (g), and the coefficient of kinetic friction (µ) to calculate the final acceleration value.