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Cement Hopper

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maths question, MEI Mechanics 1?

A crane is used to lift a hopper full of cement to a height of 20m on a building site. the hopper has mass 200kg and the cement 500kg. initially the hopper accelerates upwards at 0.05m/s squared, then it travels at constant speed for some time before decelerating at 0.1m/s squared until it is at rest. the hopper is then emptied.

i) find the tension in the crane's cable during each of the three phases of the motion and after emptying.

the cable's maximum safe load is 10,000 N.
ii) what is the greatest mass of cement that can safely be transported in the same manner?

Let T = tension in the cable (N)
Let a = acceleration of the hopper (m/s^2)
Let g = acceleration due to gravity (m/s^2)
Let m = mass (kg)

i) As the hopper accelerates upwards at 0.05m/s^2 :

T - mg = ma
T = ma + mg
T = m(a + g)
T = 700 (0.05 + 9.8)
T = 6895 N

As it travels at a constant speed, a = 0 :
So T = mg
T = (700)(9.8)
T = 6860 N

As it decelerates, a = -0.1m/s^2 :
So, T = ma + mg
T = m(a + g)
T = 700(-0.1 + 9.8)m/s^2
T = 6790 N

When the hopper is empty, and at rest, m = 200 kg, a = 0 :
So, T = mg
T = (200)(9.8)
T = 1960 N

The tension in the crane's cable during each of the three phases of the motion
and after emptying is:

As the hopper accelerates upwards, the crane's cable tension = 6895 N
As hopper travels at a constant speed, the crane's cable tension = 6860 N
As hopper decelerates, the crane's cable tension = 6790 N
When the hopper is empty, and at rest the crane's cable tension = 1960 N

(ii) Let m be the greatest mass (cement + hopper) that can be safely transported.
Since the greatest tension happens in the initial phase of acceleration at 0.05 m/s^2,
use T = ma + mg, solve for m :

T = ma + mg
T = m(a + g)
10000 = m(0.05 + 9.8)
10000 = m(9.85)
m = 10000/9.85
m = 1015 kg

Mass of the cement that can be transported
= 1015 - 200
= 815 kg

Therefore, the greatest mass of cement that can be safety transported is 815 kg.

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NEW NIP Vtg HO Life Like Erie 21100 Cement Hopper Freight Car Model Railroad
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WALTHERS 932 25372 TRINITY 100 TON CEMENT COVERED HOPPER 2PK Burlington Northern
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Tyco HO Operating Cement Hopper Car Monon CIL
Tyco HO Operating Cement Hopper Car Monon CIL
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Custom ZERBA Bramos 33720 Czech Zementsilo Cement Hopper Car CSD 2 Rail HO
Custom ZERBA Bramos 33720 Czech Zementsilo Cement Hopper Car CSD 2 Rail HO
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HO Scale Burlington Route CBQ 85124 Covered 2 bay Cement Hopper Train
HO Scale Burlington Route CBQ 85124 Covered 2 bay Cement Hopper Train
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2 Walthers Gold Line 932 5382 Cement Covered  932 7412 Twin Hopper cars NIB
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336 Tyco HO Scale Cement Hopper Car 40 Foot Monon
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Walthers HO Scale Trinity 100 Ton Cement Covered Hopper 2pack BN NIB
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LIFE LIKE Cement Hopper Car ERIE New Old Stock NIB Train Car 8514 HO Scale 4C
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Walthers decals HO Freight Western Maryland cement hopper black M71
Walthers decals HO Freight Western Maryland cement hopper black M71
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Walthers HO Scale UP Trinity 100 Ton Cement Covered Hopper
Walthers HO Scale UP Trinity 100 Ton Cement Covered Hopper
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FRATESCHI CEMENT HOPPERS WITH KNUCKLE COUPLERS
FRATESCHI CEMENT HOPPERS WITH KNUCKLE COUPLERS
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Walthers HO Scale Trinity 100 Ton Cement Covered Hopper 2pack CHTT NIB
Walthers HO Scale Trinity 100 Ton Cement Covered Hopper 2pack CHTT NIB
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Walthers HO Scale Trinity 100 Ton Cement Covered Hopper 2pack NW NIB
Walthers HO Scale Trinity 100 Ton Cement Covered Hopper 2pack NW NIB
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HO 100 ton UNION PACIFIC 2 BAY covered hopper WALTHERS cement light weathering
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HO MDC Roundhouse 2 bay Cement Hopper CONRAIL 70 ton graffiti LOOK
HO MDC Roundhouse 2 bay Cement Hopper CONRAIL 70 ton graffiti LOOK
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Life Like HO Cement Hopper Campbell Beans 8427
Life Like HO Cement Hopper Campbell Beans 8427
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Your Choice Walthers 100 Ton Cement Covered Hoppers
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Plano 086 Walkways for PS2 2 Bay Cement Hopper Walthers Car Morton Pattern
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HO SCALE MARX HURON PORTLAND CEMENT 2 BAY COVERED HOPPER CAR
HO SCALE MARX HURON PORTLAND CEMENT 2 BAY COVERED HOPPER CAR
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Life Like HO Scale Item 08530 Cement Hopper Car CO Chesapeake  Ohio
Life Like HO Scale Item 08530 Cement Hopper Car CO Chesapeake Ohio
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WALTHERS GOLDLINE HO R T R MKT TRINITY 100 TON CEMENT COVERED HOPPER 543
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Vintage Life Like Train CAMPBELLS BEANS CEMENT HOPPER Original Box 8427 HO Scale
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BOWSER 55809 Missouri Portland Cement ACF 70 Ton Covered Hopper 304 Weath1 87
BOWSER 55809 Missouri Portland Cement ACF 70 Ton Covered Hopper 304 Weath1 87
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BOWSER 56877 Dewey Portland Cement ACF 70 Ton 2 Bay CovHopper 204 Weath1 87
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WALTHERS 5404 CNW 100 Ton Cement CovrdHopper 169238 Weathered HO 1 87
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WALTHERS 5401 UP 100 Ton Cement CovrdHopper 218268 Weathered HO 1 87
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Life Like HO Scale Campbells Cement Hopper 8427 NIB
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Kadee HO Great Northern GN 34 PS 2 Cov Hopper Cement Gray 71407
Kadee HO Great Northern GN 34 PS 2 Cov Hopper Cement Gray 71407
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Kadee HO Great Northern GN 34 PS 2 Cov Hopper Cement Gray 71501
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Oddballs decals HO ATSF 2006cuft cement hopper ptd mineral red C10
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Vintage HO 51CS Box SL Cattle Gondola Tanker Cement Hopper Cars
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Walthers CSX Trinity 100T Cement Covered Hopper 2 pack RTR NIB
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